£ 


Irving  Stringham 


© 


47   J 

■ 


>LsV\s 


30° 

\MV   ■ 

In    n 

1 

W* 

i 

0 

fS  \  I 

Oo 

^. 


2fs 


Q   y   Oo    -      / 

bo 

Oo-/-/  -    Oo 

O*?  -'-.Oo 
OoxOoz  O 


&> 


-  I 


Oo  -o>  ^  O 


t 


Oo 


t**X%  i  b  to***  ~ I 


WENTWORTH'S 
SERIES    OF    MATHEMATICS. 


First  Steps  in  Number. 

Primary  Arithmetic. 

Grammar  School  Arithmetic 

High  School  Arithmetic. 

Exercises  in  Arithmetic. 

Shorter  Course  in  Algebra. 

Elements  of  Algebra.  Complete  Algebra. 

^College  Algebra.  Exercises  in  Algebra. 

Plane  Geometry. 
'"'Plane  and  Solid  Geometry. 

Exercises  in  Geometry. 
VPI.  and  Sol.  Geometry  and  PI.  Trigonometry. 

Plane  Trigonometry  and  Tables. 

Plane  and  Spherical  Trigonometry. 

purveying. 
Vpi.  and  Sph.  Trigonometry,  Surveying,  and  Tables. 

Trigonometry,  Surveying,  and  Navigation. 

Trigonometry  Formulas. 

Logarithmic  and  Trigonometric  Tables  (Seven). 

Log.  and  Trig.  Tables  (Complete  Edition). 

Analytic  Geometry. 


Special  Terms  and  Circular  on  Application. 


Plane  Trigonometry 


by 
Qt,  A.  WENTWOKTH,   A.M., 

PROFESSOR   OF   MATHEMATICS   IN   PHILLIPS   EXETER    A.CADEMY 


BOSTON,   U.S.A.: 

GINN  &  COMPANY,   PUBLISHEKS. 

1891. 


GK533 


C5*^ 


Entered,  according  to  Act  of  Congress,  in  the  year  1883,  by 

G.   A.   WENT  WORTH, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


Typography  by  J.  S.  Cushing  &  Co.,  Boston,  U.'S.A. 


Prbssworx  by  Ginn  &  Co.,  Boston,  U.S.A. 


PREFACE. 


IN  preparing  this  work  the  aim  has  been  to  furnish  just  so  much 
of  Trigonometry  as  is  actually  taught  in  our  best  schools  and 
colleges.  Consequently,  development  of  functions  in  series  and  all 
other  investigations  that  are  important  only  for  the  special  student 
have  been  omitted.  The  principles  have  been  unfolded  with  the 
utmost  brevity  consistent  with  simplicity  and  clearness,  and  inter- 
esting problems  have  been  selected  with  a  view  to  awaken  a  real 
love  for  the  study.  Much  time  and  labor  have  been  spent  in  devis- 
ing the  simplest  proofs  for  the  propositions,  and  in  exhibiting  the 
best  methods  of  arranging  the  logarithmic  work. 

The  object  of  the  work  on  Surveying  and  Navigation  is  to  pre- 
sent these  subjects  in  a  clear  and  intelligible  way,  according  to  the 
best  methods  in  actual  use  ;  and  also  to  present  them  in  so  small  a 
compass  that  students  in  general  may  find  the  time  to  acquire  a 
competent  knowledge  of  these  very  interesting  and  important  studies. 

The  author  is  under  particular  obligation  for  assistance  to  G.  A.  Hill, 
A.M.,  of  Cambridge,  Mass.,  and  to  Prof.  James  L.  Patterson,  of  Law- 
rencerille,  N.J. 

G.  A.  WENTWORTH. 
Phillips  Exeter  Academy, 
September,  1882. 


800564 


CONTENTS. 


PLANE  TRIGONOMETRY. 

CHAPTER  I.     Functions  of  Acute  Angles: 

Definitions,  1 ;  representation  of  functions  by  lines,  7 ;  changes  in 
the  functions  as  the  angle  changes,  9 ;  functions  of  complementary- 
angles,  10 ;  relations  of  the  functions  of  an  angle,  11 ;  formulas  for 
finding  all  the  other  functions  of  an  angle  when  one  function  of  the 
angle  is  given,  13;  functions  of  45°,  30°,  60°,  15. 

CHAPTER   II.     The  Right  Triangle: 

Solution:  Case  I.,  when  an  acute  angle  and  the  hypotenuse  are 
given,  16;  Case  II.,  when  an  acute  angle  and  the  opposite  leg  are 
given,  17;  Case  III.,  when  an  acute  angle  and  the  adjacent  leg 
are  given,  17 ;  Case  IV.,  when  the  hypotenuse  and  a  leg  are  given, 
18 ;  Case  V.,  when  the  two  legs  are  given,  18 ;  general  method  of 
solving  the  right  triangle,  19 ;  area  of  the  right  triangle,  20 ;  the 
isosceles  triangle,  24;  the  regular  polygon,  26. 

CHAPTER  III.    Goniometry: 

Definition  of  Goniometry,  28;  angles  of  any  magnitude,  28;  gen- 
eral definitions  of  the  functions  of  angles,  29 ;  algebraic  signs  of  the 
functions,  31 ;  functions  of  a  variable  angle,  32  ;  functions  of  angles 
larger  than  360°,  34 ;  formulas  for  acute  angles  extended  to  all  angles, 
35  ;  reduction  of  the  functions  of  all  angles  to  the  functions  of  angles 
in  the  first  quadrant,  38  ;  functions  of  angles  that  differ  by  90°,  40 ; 
functions  of  a  negative  angle,  41 ;  functions  of  the  sum  of  two  angles, 
43 ;  functions  of  the  difference  of  two  angles,  45 ;  functions  of  twice 
an  angle,  47  ;  functions  of  half  an  angle,  47 ;  sums  and  differences  of 
functions,  48.  > 


VI  TRIGONOMETRY. 


CHAPTER  IV.     The  Oblique  Triangle: 

Law  of  sines,  50  ;  law  of  cosines,  52 ;  law  of  tangents,  52.  Solu- 
tion :  Case  I.,  when  one  side  and  two  angles  are  given,  54  ;  Case  II., 
when  two  sides  and  the  angle  opposite  to  one  of  them  are  given,  56 ; 
Case  III.,  when  two  sides  and  the  included  angle  are  given,  60 ;  Case 
IV.,  when  the  three  sides  are  given,  64  ;  area  of  a  triangle,  68  ;  mis- 
cellaneous problems,  71-88. 

EXAMINATION  PAPERS,  89-102. 


SPHERICAL  TRIGONOMETRY. 

CHAPTER  V.    The  Right  Spherical  Triangle: 

Introduction,  103  ;  formulas  relating  to  right  spherical  triangles, 
105 ;  Napier's  rules,  108.  Solution  :  Case  I.,  when  the  two  legs  are 
given,  110  ;  Case  II.,  when  the  hypotenuse  and  a  leg  are  given,  110 ; 
Case  III.,  when  a  leg  and  the  opposite  angle  are  given,  111 ;  Case  IV., 
when  a  leg  and  an  adjacent  angle  are  given,  111 ;  Case  V.,  when  the 
hypotenuse  and  an  oblique  angle  are  given,  111 ;  Case  VI.,  when 
the  two  oblique  angles  are  given,  111  ;  solution  of  the  isosceles 
spherical  triangle,  116  ;  solution  of  a  regular  spherical  polygon,  116. 

CHAPTER  VI.     The  Oblique  Spherical  Triangle: 

Fundamental  formulas,  117  ;  formulas  for  half  angles  and  sides, 
119 ;  Gauss's  equations  and  Napier's  analogies,  121.  Solution :  Case  I., 
when  two  sides  and  the  included  angle  are  given,  123  ;  Case  II.,  when 
two  angles  and  the  included  side  are  given,  125  ;  Case  III.,  when  two 
sides  and  an  angle  opposite  to  one  of  them  are  given,  127  ;  Case  IV., 
when  two  angles  and  a  side  opposite  to  one  of  them  are  given,  129  ; 
Case  V.,  when  the  three  sides  are  given,  130;  Case  VI.,  when  the 
three  angles  are  given,  131  ;  area  of  a  spherical  triangle,  133. 

CHAPTER  VII.    Applications  of  Spherical  Trigonometry  : 

Problem,  to  reduce  an  angle  measured  in  space  to  the  horizon,  136  •, 
problem,  to  find  the  distance  between  two  places  on  the  earth's  sur- 
face when  the  latitudes  of  the  places  and  the  difference  of  their  longi- 
tudes are  known,  137 ;  the  celestial  sphere,  137 ;  spherical  co-ordinates, 
140 ;  the  astronomical  triangle,  142;  astronomical  problems,  143-146. 


PLANE   TRIGONOMETRY. 


CHAPTER   I. 


TRIGONOMETRIC    FUNCTIONS    OF    ACUTE    ANGLES. 


§  1.   Definitions. 

The  sides  and  angles  of  a  plane  triangle  are  so  related  that 
any  three  given  parts,  provided  at  least  one  of  them  is  a  side, 
determine  the  shape  and  the  size  of  the  triangle. 

Geometry  shows  how,  from  three  such  parts,  to  construct  the 
triangle  and  find  the  values  of  the  unknown  parts. 

Trigonometry  shows  how  to  compute  the  unknown  parts  of  a 
triangle  from  the  numerical  values  of  the  given  parts. 

Geometry  shows  in  a  general  way  that  the  sides  and  angles 
of  a  triangle  are  mutually  dependent.  Trigonometry  begins 
by  showing  the  exact  nature  of  this  dependence  in  the  right 
triangle,  and  for  this  purpose  employs  the  ratios  of  its  sides. 

Let  MAN (Fig.  1)  be  an  acute  angle.     If  from  any  points 

B,  D,  F, in  one  of  its  sides 

perpendiculars  BC,  DF,  FG, 

are  let  fall  to  the  other  side,  then 
the  right  triangles  ABC,  ADE, 

AFG, thus  formed  have  the 

angle  A  common,  and  are  there- 
fore mutually  equiangular  and 
similar.  Hence,  the  ratios  of 
their  corresponding  sides,  pair  by 
pair,  are  equal.     That  is, 


AC _AE _AG , 
AB     AD     AF' 


AC^AE^AG 
BC     DF     FG 


;  etc. 


TRIGONOMETRY. 


Hence,  for  every  value  of  an  acute  angle  A  there  are  certain 
numbers  that  express  the  values  of  the  ratios  of  the  sides  in 
all  right  triangles  that  have  this  acute  angle  A. 
There  are  altogether  six  different  ratios : 

I.  The  ratio  of  the  opposite  leg  to  the  hypotenuse  is  called 
the  Sine  of  A,  and  is  written  sin  A. 

II.  The  ratio  of  the  adjacent  leg  to  the  hypotenuse  is  called 
the  Cosine  of  A,  and  written  cos  A 

III.  The  ratio  of  the  opposite  leg  to  the  adjacent  leg  is 
called  the  Tangent  of  A,  and  written  tan  A 

IV.  The  ratio  of  the  adjacent  leg  to  the  opposite  leg  is 
called  the  Cotangent  of  A,  and  written  cot  A. 

V.  The  ratio  of  the  hypotenuse  to  the  adjacent  leg  is  called 
the  /Secant  of  A,  and  written  sec  A. 

VI.  The  ratio  of  the  hypotenuse  to  the  opposite  leg  is  called 
the  Cosecant  of  A.  and  written  esc  A. 


In  the  right  triangle  ABC 
(Fig.  2)  let  a,  b,  c  denote  the 
lengths  of  the  sides  opposite  to 
the  acute  angles  A,  B,  and  the 
right  angle  C,  respectively,  these 
lengths  being  all  expressed  in 
terms  of  a  common  unit.    Then, 


A      a      opposite  leg 

sin  A  —  -  =  ^ — ; -• 

c       hypotenuse 


. b  _  adjacent  leg 

c       hypotenuse 


tan..l-a-°PP°siteleg 
b      adjacent  leg 


.    .       b      adjacent  leg 

cot.  A  =  -  =  — - — ^— -j-^' 

a      opposite  leg 


sec  A 


hypotc 


b      adjacent  leg 


esc  A  = 


c hypotenuse 

a      opposite  leg 


These  six  ratios  are  called  the  Trigonometric  Functions  of  the 
angle  A. 


TRIGONOMETRIC    FUNCTIONS. 


Exercise  I. 

1.  What  are  the  functions  of  the  other  acute  angle  B  of  the 
triangle  ABC  (Fig.  2)? 

2.  Prove  that  if  two  angles,  A  and  B,  are  complements  of 
each  other  (i.e.,  ifA  +  B  =  90°),  then, 

sin  A  =  cos  B,     tan  A  =  cot  B,     sec  A  =  esc  B ; 
cos  A  =  sin  B,     cot  A  =  tan  B,     esc  ^4  =  sec  B. 

3.  Find  the  values  of  the  functions  of  A,  if  a,  b,  c  respec- 
tively have  the  following  values : 

(i.)   3,     4,     5.      (iii.)   8,  15,  17.       (v.)   3.9,     8,        8.9. 
(ii.)   5,  12,  13.      (iv.)   9,  40,  41.      (vi.)   1.19,  1.20,  1.69. 

4.  What  condition  must  be  fulfilled  by  the  lengths  of  the 
three  lines  a,  b,  c  (Fig.  2)  in  order  to  make  them  the  sides  of 
a  right  triangle  ?    Is  this  condition  fulfilled  in  Example  3  ? 

5.  Find  the  values  of  the  functions  of  A,  if  a,  b}  c  respec- 
tively have  the  following  values  : 

(i.)    2  raw,  m2—n2,  m2-\~n2.  (iii.)  pqr,  qrs,  rsp, 

(n.)   £-,  x  +  y,  --OL.  (iv.)   — ,  — ,  — . 

x  —  y  %  —  y  pq     sq    ps 

6.  Prove  that  the  values  of  a,  b,  c,  in  (i.)  and  (ii.),  Example 
5,  satisfy  the  condition  necessary  to  make  them  the  sides  of 
a  right  triangle. 

7.  What  equations  of  condition  must  be  satisfied  by  the 
values  of  a,  b,  c,  in  (iii.)  and  (iv.),  Example  5,  in  order  that  the 
values  may  represent  the  sides  of  a  right  triangle? 

Compute  the  functions  of  A  and  B  when, 

8.  a  =  24,   5  =  143.  11.  a  =  ^p2  +  q2,    b  =  V2pq. 

9.  a  =  0.264,   c  =  0.265.       12.  a  =  Vp2+pq,    c  =  p  +  q. 
10.   5  =  9.5,   ,  =  19.3.  13    b==2Vfi,  c=p  +  q. 


6 


TRIGONOMETRY. 


Compute  the  functions  of  A  when, 

14.  a  =  2b.  16.  a  +  b  = 

15.  a=t<r.  17.  a-b  =  - 

18.  Find  a  if  sin  A  =  f  and  c  =  20.5. 

19.  Find  6  if  cos  ^4  =  0.44  and  c  =  3.5. 

20.  Find  a  if  tan  A  =  ^  and  b  =  2T5T. 

21.  Find  6  if  cot^4  =  4  and  a  =  17. 

22.  Find  c  if  sec  A  =  2  and  5  ==  20. 

23.  Find  c  if  esc  J.  =  6.45  and  a  =  35.6. 
Construct  a  right  triangle  •  given, 

24.  c  =  6,   tan^=f.  26.   b  =  2, 

25.  a=3.5,   cos  .4= -J.  27.   5  =  4, 

28.  In  a  right  triangle,  £  =  2.5  miles,  sin^l  =  0.6,  cos  A  = 
0.8 ;  compute  the  legs. 

29.  Construct  (with  a  protractor)  the  A  20°,  40°,  and  70°; 
determine  their  functions  by  measuring  the  necessary  lines, 
and  compare  the  values  obtained  in  this  way  with  the  more 
correct  values  given  in  the  following  table  : 


sin  A  =  0.6. 
esc  A  =4. 


20° 
40° 
70° 

sin 

cos 

tan 

cot 

sec 

CSC 

0.342 
0.643 
0.940 

0.940 
0.766 
0.342 

0.364 
0.839 

2.747 

2.747 
1.192 
0.364 

1.064 
1.305 
2.924 

2.924 
1.556 
1.064 

30.  Find,  by  means  of  the  above  table,  the  legs  of  a  right 
triangle  if  A  »  20°,  e  =  1 ;  also,  if  A  =  20°,  c  ±=  4. 

31.  In  a  right  triangle,  given  a  =  3  and  c  =  5;  find  the 
hypotenuse  of  a  similar  triangle  in  which  a  =  240,000  miles. 

32.  By  dividing  the  length  of  a  vertical  rod  by  the  length 
of  its  horizontal  shadow,  the  tangent  of  the  angle  of  elevation 
of  the  sun  at  the  time  of  observation  was  found  to  be  0.82. 
How  high  is  a  tower,  if  the  length  of  its  horizontal  shadow  at 
the  same  time  is  174.3  yards? 


TRIGONOMETRIC    FUNCTIONS. 


§  2.   Representation  of  Functions  by  Lines. 

The  functions  of  an  angle,  being  ratios,  are  numbers;  but 
we  may  represent  them  by  lines  if  we  first  choose  a  unit  of 
length,  and  then  construct  right  triangles,  such  that  the  de- 
nominators of  the  ratios  shall  be  equal  to  this  unit.  The  most 
convenient  way  to  do  this  is  as  follows  : 

About  a  point  0  (Fig.  3)  as  a 
centre,  with  a  radius  equal  to  one 
unit  of  length,  describe  a  circle 
and  draw  two  diameters  A  A'  and 
BB'  perpendicular  to  each  other. 

The  circle  with  radius  equal  to 
1  is  called  a  unit  circle,  A  A'  the 
horizontal,  and  BB'  the  vertical 
diameter. 

Let  AOP  be  an  acute  angle, 
and  let  its  value  (in  degrees,  etc.) 
be  denoted  by  x.    We  may  regard 

the  Z  x  as  generated  by  a  radius  OP  that  revolves  about  0 
from  the  position  OA  to  the  position  shown  in  the  figure ; 
viewed  in  this  way,  OP  is  called  the  moving  radius. 

Draw  PM  _L  to  OA.  In  the  rt.  A  OPM  the  hypotenuse 
OP=  1 ;  therefore,  sin  a;  =  PM;  cos#  =  OM. 

Since  PM  is  equal  to  ON,  and  OiVis  the  projection  of  OP 
on  BB',  and  since  OM  is  the  projection  of  OP  on  AA! ,  there- 
fore, in  a  unit  circle, 

sin x  =  projection  of  moving  radius  on  vertical  diameter ; 

cos  a;  =  projection  of  moving  radius  on  horizontal  diameter. 

Through  A  and  B  draw  tangents  to  the  circle  meeting  OP, 
produced  in  T  and  8,  respectively ;  then,  in  the  rt.  A  OA  T, 
the  leg  OA  =  1,  and  in  the  rt.  A  OBS,  the  leg  0B  =  1;  while 
the  Z  OSB  =  Z  x  (why  ?).     Therefore, 


tan#  =  A T;  seox  —  OT;  cotx  =  BS;  cscx  =  OS. 


TRIGONOMETRY. 


These  six  line  values  (as  they  may  be  termed)  of  the  func- 
tions are  all  expressed  in  terms  of  the  radius  of  the  circle  as  a 
unit ;  and  it  is  clear  that  as  the  angle  varies  in  value  the  line 
values  of  the  functions  will  always  remain  equal  numerically 
to  the  ratio  values.  Hence,  in  studying  the  changes  in  the 
functions  as  the  angle  is  supposed  to  vary,  we  may  employ  the 
simpler  line  values  instead  of  the  ratio  values. 

Exercise  II. 

1.  Represent  by  lines  the  functions  of  a  larger  angle  than 
that  shown  in  Fig.  3. 

2.  Show  that  sin x  is  less  than  tana;. 

3.  Show  that  sec x  is  greater  than  tana;. 

4.  Show  that  esc  x  is  greater  than  cot  x. 
Construct  the  angle  x  if, 

5.  tana;  =  3.  7.    cosa;  =  -|-.  9.   sina;  =  2cosa;. 

6.  esc  x  =  2.  8.   sina;  =  cosa;.         10.   4  sina;  =  tana;. 

11.  Show  that  the  sine  of  an  angle  is  equal  to  one-half  the 
chord  of  twice  the  angle. 

12.  Find  x  if  sin  a;  is  equal  to  one-half  the  side  of  a  regular 
inscribed  decagon. 

13.  Given  x  and  y  (x-\-y  being  less  than  90°)  ;  construct  the 
value  of  sin  (x  +  y)  —  sin  x. 

14.  Given  x  and  y  {x-\-y  being  less  than  90°) ;  construct  the 
value  of  tan  (x  -j-  y)  —  sin  (x-\-y)-\-  tana;  —  sin  a;. 

Given  an  angle  x ;  construct  an  angle  y  such  that, 

15.  siny  =  2 sina;.  17.    tany  ==  3 tana;. 

16.  cosy  —  \  cos x.  18.   secy  =  csca;. 

19.  Show  by  construction  that  2  sin  A  >  sin  2  A. 

20.  Given  two  angles  A  and  B  (A  +  B  being  less  than  90°) ; 
show  that  sin  (A  -f  B)  <  sin  A  -f  sin  B. 

21.  Given  sina;  in  a  unit  circle;  find  the  length  of  a  line 
corresponding  in  position  to  sina;  in  a  circle -whose  radius  is  r  ? 

22.  In  a  right  triangle,  given  the  hypotenuse  c,  and  also 
sin  A  =  m,  cos  A  =  n ;  find  the  legs. 


TRIGONOMETRIC    FUNCTIONS. 


§  3.  Changes  in  the  Functions  as  the  Angle  Changes. 

If  we  suppose  the  Z  AOP,  or  x  (Fig.  4)  to  increase  gradtu 
ally  by  the  revolution  of  the  moving 
radius  OP  about  0,  the  point  P  will 
move  along  the  arc  AB  towards  B, 
T  will  move  along  the  tangent  A  T 
away  from  A,  S  will  move  along 
the  tangent  BS  towards  B,  and  M 
will  move  along  the  radius  OA 
towards  0. 

Hence,  the  lines  PM,  AT,  OT 
will  gradually  increase  in  length, 
and  the  lines  OM,  BS,  OS,  will 
gradually  decrease.     That  is, 

As  an  acute  angle  increases,  its 
sine,  idngcnt,  and  secant  also  in- 
crease, while  its  cosine,  cotangent,  and  cosecant  decrease. 

On  the  other  hand,  if  we  suppose  x  to  decrease  gradually, 
the  reverse  changes  in  its  functions  will  occur. 

If  we  suppose  x  to  decrease  to  0°,  OP  will  coincide  with  OA 
and  be  parallel  to  BS.  Therefore,  PM  and  A  T  will  vanish, 
OM  will  become  equal  to  OA,  while  BS  said  OS  will  each  be 
infinitely  long,  and  be  represented  in  value  by  the  symbol  oo. 

And  if  we  suppose  x  to  increase  to  90°,  OP  will  coincide 
with  OB  and  be  parallel  to  AT.  Therefore,  PM  and  OS  will 
each  be  equal  to  OB,  OM  and  BS  will  vanish,  while  AT  and 
0 Twill  each  be  infinite  in  length. 

Hence,  as  the  angle  x  increases  from  0°  to  90°, 

sin  x  increases  from  0  to  1, 
cos  x  decreases  from  1  to  0, 
tana;  increases  from  0  to  oo, 
cot  x  decreases  from  oo  to  0, 
sec  x  increases  from  1  to  cos 
esc  x  decreases  from  oo  to  1. 


10 


TRIGONOMETRY. 


The  values  of  the  functions  of  0°  and  of  90°  are  the  limiting 
values  of  the  functions  of  an  acute  angle.  It  is  evident  that 
(disregarding  the  limiting  values), 

Sines  and  cosines  are  always  less  than  1 ; 

Secants  and  cosecants  are  always  greater  than  1 ; 

Tangents  and  cotangents  have  all  values  between  0  and  oo. 

Remabk.  We  are  now  able  to  understand  why  the  sine,  cosine,  etc., 
of  an  angle  are  called  functions  of  the  angle.  By  &  function  of  any  mag- 
nitude is  meant  another  magnitude  which  remains  the  same  so  long  as 
the  first  magnitude  remains  the  same,  but  changes  in  value  for  every 
change  in  the  value  of  the  first  magnitude.  This,  as  we  now  see,  is  the 
relation  in  which  the  sine,  cosine,  etc.,  of  an  angle  stand  to  the  angle. 


§4.   Functions  of  Complementary  Angles. 

The  general  form  of  two  complementary  angles  is  A  and 
90°  -  A. 

In  the  rt.  A  ABO  (Fig.  5) 
A  +  B  =  90°,  hence  B  =  90°  -  A. 
Therefore  (§  1), 

sin  A  =  cos  B  =  cos  (90°  -  A), 
cos  A  =  sin  B  =  sin  (90°  —  A), 
tan  A  =  cot  B  =  cot  (90°  -  A), 
cot  A  =  tan  B  =  tan  (90°  -  A), 
sec  A  =  esc  B  =  esc  (90°  —  A), 
esc  A  =  sec  B  =  sec  (90°  -  A). 
Therefore, 

■    Each  function  of  an  acute  angle  is  equal  to  the  co-named 

function  of  the  complementary  angle. 

Note.  Cosine,  cotangent,  and  cosecant  are  sometimes  called  co-functions; 
the  words  are  simply  abbreviated  forms  of  complement's  sine,  complement s 
tangent,  and  complement's  secant. 

Hence,  also, 

Any  function  of  an  angle  between  45°  and  90°  may  be  found 
by  taking  the  co-named  function  of  the  complementary  angle 
between  0°  and  45°. 


TRIGONOMETRIC    FUNCTIONS.  11 


Exercise  III. 

1.  Express  the  following  functions  as  functions  of  the  com- 
plementary angle : 

sin  30°.         tan  89°.         esc  18°  10'.         cot  82°  19'. 
cos 45°.        cot  15°.         cos 37°  24'.         esc 54°  46'. 

2.  Express  the  following  functions  as  functions  of  an  angle 
less  than  45°  : 

sin60°.         tan 57°.         csc69°2\  cot89°  59'. 

cos75°.        cot84°.         cos85°39'.         esc  45°  1'. 

3.  Given  tan  30°  =  i  V3  ;  find  cot  60°. 

4.  Given  tan  A  =  cot  A  ;  find  A:- " 

5.  Given  cos  A  =  sin  2  A  ;  find  A. 

6.  Given  sin  A  =  cos  2  A  ;  find  A. 

7.  Given  cos  A  =  sin  (45°  —  hA);  find  A. 

8.  Given  cot  i  A  —  tan  A ;  find  A. 

9.  Given  tan  (45°  +  A)  =  cot  A  ;  find  A. 

10.  Find  A  if  sin  A  =  cos  4-4. 

11.  Find  A  if  cot  A  =  tan  8  A. 

12.  Find  A  if  cot  A  =  tan  nA. 

§  5.    Relations  of  the  Functions  of  an  Angle. 
Since  (Fig.  5)  a2  -\-  b2  =  c2,  therefore, 

a2  ,  62      .  fa\*     fb^2 


Therefore  (§  1),    (sin  ^)2  +  (cos  ^L)2  =  1 ; 

or,  as  usually  written  for  convenience, 

sin2A  +  cos2A  =  l.  [1] 

That  is :    The  sum  of  the  squares  of  the  sine  and  the  cosine  of 
an  angle  is  equal  to  unity. 


12  TRIGONOMETRY. 


Formula  [1]  enables  us  to  find  the  cosine  of  an  angle  when 
the  sine  is  known,  and  vice  versa.  The  values  of  sin  A  and  of 
cos  A  deduced  from  [1]  are  : 


sin 


A  =  Vl  —  cosM,     cos  A  ==  Vl  —  sin2  A 


a.  abaca 

Since  — j —  =  _-x7  =  T, 

c      c      c      b      b 

therefore  (§  1),  tan  A  =  —£  [2] 

That  is :  The  tangent  of  an  angle  is  equal  to  the  sine  divided 
by  the  cosine. 

Formula  [2]  enables  us  to  find  the  tangent  of  an  angle  when 
the  sine  and  the  cosine  are  known. 


a .    c 


b      c      ~  ,     a      b 

Since      -X-  —  1,      -XT  =  1,    and     TX- 

c      o  b      a 


therefore  (§  1),  sin  A  X  esc  A  =  1 1 

cos  A  x  sec  A  =  1  I  •    -  [3] 

tan  A  X  cot  A  =  1 J 

That  is :  The  sine  and  the  cosecant  of  an  angle,  the  cosine 
and  secant,  and  the  tangent  and  cotangent,  pair  by  pair,  are 
reciprocals. 

The  equations  in  [3]  enable  us  to  find  an  unknown  function 
contained  in  any  pair  of  these  reciprocals  when  the  other  func- 
tion in  this  pair  is  known. 

Exercise  IV. 

1.  Prove  Formulas  [l]--[3],  using  for  the  functions  the  line 
Values  in  unit  circle  given  in  §  3. 

2.  Prove  that  1  -f-  tan2^.  ==  sec2  A. 

Hint.   Divide  the  terms  of  the  equation  a2  +  6*  —  c2  by  b*. 

3.  Prove  that  l  +  cotM  =  csc2X  t 

.     _  .  cos  ^4 

4.  Prove  that  cot  A  =  - — f 

sin  A 


TRIGONOMETRIC   FUNCTIONS.  13 


§6.   Application  of  Formulas  [l]-[3]. 

Formulas  [1],  [2],  and  [3]  enable  us,  when  any  one  function 
of  an  angle  is  given,  to  find  all  the  others.  A  given  value  of 
any  one  function,  therefore,  determines  all  the  others. 

Example  1.    Given  sin  A  =  -§  ;  find  the  other  functions. 

By  [1],  cos  A  =  Vr=l-  =  V|  =  JV5. 

By  [2],         ta^  =  !-4v5  =  fx-|=A. 

By  |"3l,  cot  A  =  ——.    secA  = ,    csc.4  =  — 

Example  2.  .  Given  tan  A  —  3  ;  find  the  other  functions. 

By  [2],  si^4  =  3. 

cos  A 

And  by  [1],  sin2 A  -f-  cos2^.  =  1. 

If  we  solve  these  equations  (regarding  sin  A  and  cos  A  as 
two  unknown  quantities),  we  find  that, 

sin  A  =  3  VTV     cos  A  =  VTV 

Then  by  [3],  cot  A  =  |,     sec  ^4  =  VIO,     esc  J.  =  -J  VlO, 

Example  3.    Given  sec  A  =  m;  find  the  other  functions. 

1 


By  [3],  cos  A  = 


m 


By[l],  sin^  =  Jl-i  =  J^i=iv: 


%  [2],  [3],    tan  A  =  Vm2  -  1 ,       cot  ^  -  — ^ 
esc  -4 


V- 


III- 


14  TRIGONOMETRY. 


Exercise  V. 
Find  the  values  of  the  other  functions  when, 

1.  sin  A  =  |f.        ©    tan^4  =  |.  9.    csc^4=V2 

2.  sin  A  =  0.8.        6.    cot  A  =  1.         10.    sin.4  =  m. 

3.  cos  ^4  =  ff         7.   cot  A  =  0.5.      11.   sin  ^4  = 


1  +  ra2 


4.    cos,4  =  0.28.      8.    sec  A  =  2.  2mi 

12.    cos  ^4  =  — - — • 
raa-fn 

13.  Given  tan  45°  =  1  ;  find  the  other  functions  of  45°. 

14.  Given  sin  30°  =  -J- ;  find  the  other  functions  of  30°. 

15.  Given  esc  60°  =  -§  V3  ;  find  the  other  functions  of  60°. 

16.  Given  tan  15°  =  2  -  V3  ;  find  the  other  functions  of  15°. 

17.  Given  cot  22°  30'=  V2  +  1;    find  the  other  functions 
of  22°  30'. 

18.  Given  sin  0°  =  0 ;  find  the  other  functions  of  0°. 

19.  Given  sin  90°  =  1 ;  find  the  other  functions  of  90°. 

20.  Given  tan  90°  =  co  ;  find  the  other  functions  of  90°. 

21.  Express  the  values  of  all  the  other  functions  in  terms 
of  sin  A. 

22.  Express  the  values  of  all  the  other  functions  in  terms 
of  cos  A. 

23.  Express  the  values  of  all  the  other  functions  in  terms 
of  tan  A. 

24.  Express  the  values  of  all  the  other  functions  in  terms 
of  cot  A. 

25.  Given  2  sin  A  =  cos  A  ;  find  sin  A  and  cos  A. 

26.  Given  4  sin  A  =  tan  A  ;  find  sin  A  and  tan  A. 

27.  If  sin  A  :  cos  A  —  9  :  40,  find  sin  A  and  cos  A, 

28.  Transform  the  quantity  tan2 A  -f-  cot2  J.  —  sin2^4  —  cos2^4 
into  a  form  containing  only  cos  A. 

29.  Prove  that  sin  A  -f  cos  ^4  =  (1  -f-  tan  A)  cos  A. 

30.  Prove  that  tan  A  -j-  cot  ^4  =  sec  A  X  esc  A. 


TRIGONOMETRIC    FUNCTIONS. 


15 


§  7.    Functions  of  45°. 

Let  ABC  (Fig.  6)  be  an  isosceles  right  triangle,  in  which 
the  length  of  the  hypotenuse  AB 
is  equal  to  1 ;  then  AC  is  equal  to 
BC,  and  the  angle  A  is  equal  to  45°. 
Since  AC  +  BC  =  1,  therefore 
2AC2  =  1,  and^C  =  VJ  =  ^V2. 

Therefore  (§  1), 

sin  45°  =  cos45°  =  £V2. 
tan450  =  cot45°  =  l. 

sec  45°  =  esc  45°  =  V2.  j  \/2 

Fig.  6. 

§8.    Functions  of  30°  and  60°. 

Let  ABC  be  .an  equilateral  triangle  in  which  the  length 
of  each  side  is  equal  to  1 ;  and  let  CD  bisect  the  angle  C. 
Then  CD  is  perpendicular  to  AB  and  bisects  AB ;  hence,. 
AD  =  |,  and  CD  =  VF^T  =  V I  =  tV& 

In  the  right  triangle  ADC,  the  angle  ACD  =  30°,  and  the 
angle  CAD  =  60°. 

Whence  (§  1), 
sin  30°  =  cos  60°  =  i 


cos  30°  =  sin  60° 
tan  30°  =  cot  60° 

cot  30°  =  tan  60° 
sec  30°  =  esc  60° 


iV8. 

V3. 

2 


V3      ■ 
esc  30°  =  sec  60°  =  2. 

The  results  for  sine  and  cosine  of  30°,  45°,  and  60°  may  be 
easily  remembered  by  arranging  them  in  the  following  form : 


Angle .  .  . 

30° 

45° 

60° 

1  =  0.5 

Sine  .... 

} 

iV2 

iVE 

*V§  =  0.70711 

Cosine.  .  . 

iV3 

lV2 

| 

iV3  =  0.86603 
1 

CHAPTER  II. 


THE    RIGHT    TRIANGLE. 


§  9.   The  Given  Parts. 

In  order  to  solve  a  right  triangle,  two  parts  besides  the  right 
angle  must  be  given,  one  of  them  at  least  being  a  side. 
The  two  given  parts  may  be  : 

I.    An  acute  angle  and  the  hypotenuse. 
II.    An  acute  angle  and  the  opposite  leg. 

III.  An  acute  angle  and  the  adjacent  leg. 

IV.  The  hypotenuse  and  a  leg. 
V.    The  two  legs. 

§  10.   Case  I. 
Given  A  =  34°  28'  and  c  =  18.75  ;  required  B,  a,  b. 

B        1.  J5  =  90°-^4  =  55°32'. 


2.  1  =sin^ 
c 


'.  a 


=  c  sin  A. 


b 

Fig.  8. 

log  a        =  log  c  +  log  sin  A 
logc        =    1.27300 
log  sin  A=   9.75276* 

log  a        =    1.02576 
a        =  10.61 


cos  A  ;  .*.  b  =  ccosA. 


log  b         —  log  c  -f-  log  cos  A 
logc        =    1.27300 
log  cos  A=    9.91617 

log&        =    1.18917 
b        =  15.459 


*  For  Logarithms,  and  directions  how  to  use  them,  see  Wentworth 
and  Hill's  Five-place  Tables. 

When  —10  belongs  to  a  logarithm  or  cologarithm,  and  is  not  written, 
it  must  be  remembered  that  the  logarithm  or  cologarithm  is  10  too  large. 


THE    RIGHT    TRIANGLE. 


17 


§  11.   Case  II. 
Given  A  =  62*  10',  a  =  78  ;  find  B,  b,  c. 

1.  .5  =  90°-  A  =  27°  50'. 

2.  -  =  cot^4;   .'.b  —  a  cot  A 


3.  -  =  sin.4. 
c 

.'.  a  =csmA,  and 


a 


sin  ^4 


log  b        =  loga  +  logcot^4 
log  a        =    1.89209 
log  cot  A  =   9.72262 

loe&        =    1.61471 


^g 


&        =41.182 


logc 
log  a 


=  log  a  +  colog  sin  A 
===    1.89209 


colog  sin  A=    0.05340 

log*  =    1.94549 

c  =88.204 


§  12.   Case  III. 
Given  A  =  50°  2',   b  =  88 ;  find  B,  a,  c. 

1.  5  =  90°  -  ^4  =  39°  58'. 

2.  7  =  tan^4;  .'.  a  =  Man  A 
b 


8.* 


cos  .4. 


ccos^l,  and  <?  = 


cos  ^4 


log  a        as  log  5  +  log  tan  A 
log  5        =      1.94448 
log  tan  A=    10.0767 

log  a        ===     2.02118 
a        =  105.0 


logc 
log  b 


=  log  b  -f-  colog  cos  ^4 
=      1.94448 
colog  cos  A=     0.19223 

logc  =     2.13671 


=  137.0 


18 


TRIGONOMETRY. 


§  13.   Case  IV. 

Given  c  =  58.40,  a  =  47.55  ;  find 
A,  B,  b. 

1.  sini  =  — 


log  sin  A  =  log  a  -f  colog  c 
log  a        =1.67715 
colog  c     =8.23359 


log  sin  A 


9.91074 
v4  =  54°31' 
B  =  35°  29' 


2.  .£  =  90°  -A 

cot  A  ; 


3.* 

a 


b  =  acotA. 


log  b         =  log  a  +  log  cot  A 
loga        =    1.67715 
log  cot  A=    9.85300 

log6         =    1.53015 
b        =  33.896 


14.   Case  V. 

Given  a  =  40,   5  =  27;    find  A, 
B,c. 

1.  tan^4=— • 

b 

2.  ^  =  90° -A. 

n    a 


log  tan  A  —  log  a  +  colog  b 
loga         =    1.60206 
colog  b     =    8.56864 

log  tan  A  =  10.17070 
A  =  55°  59' 
.£  =  34°!' 


c 


sin  A. 


a  =csmA 


sin  ^4 

logc  =  log  a + colog  sin  A 

loga  =   1.60206 

colog  sin  A=  0.08152 


>g< 


=   1.68358 
=  48.259 


THE    RIGHT    TRIANGLE.  19 

§  15.   General  Method  of  Solving  the  Right  Triangle. 

From  these  five  cases  it  appears  that  the  general  method  of 
finding  an  unknown  part  in  a  right  triangle  is  as  follows : 

Choose  from  the  equation  A-\-  B  =  90°,  and  the  equations  that 
define  the  functions  of  the  angles,  an  equation  in  which  the  re- 
quired part  only  is  unknown;  solve  this  equation,  if  necessary, 
to  find  the  value  of  the  unknown  part ;  then  compute  the  value, 
using  logarithms  whenever  convenient. 

Note  1.  In  Cases  IV.  and  V.  the  unknown  side  may  also  be  found 
by  Geometry,  from  the  equation  a2  +  b*  =  c2 ;  whence  we  obtain 


(for  Case  IV.)        b  =  Vc^a*  =  V(c  +  a)(c-a) 


(for  Case  V.)  c  =  Va2  +  62. 

These  equations  express  the  values  of  b  and  c  directly  in  terms  of  the 
two  given  sides  ;  and  if  the  values  of  the  sides  are  simple  numbers  [e.g. 
5,  12,  13),  it  is  often  easier  to  find  b  or  c  in  this  way.  But  this  value  of 
c  is  not  adapted  to  logarithms,  and  this  value  of  b  is  not  so  readily  worked 
out  by  logarithms  as  the  value  of  b  given  in  §  13. 

Note  2.  In  Case  IV.  if  the  given  sides  (here  a  and  c)  are  nearly  alike 
in  value,  then  A  is  near  90°,  and  its  value  cannot  be  accurately  found 
from  the  tables,  because  the  sines  of  large  angles  differ  little  in  value  (as 
is  evident  from  Fig.  4).  In  this  case  it  is  better  to  find  B  first,  by  means 
of  a  formula  proved  later  (see  page  47) ;  viz., 

tan^^A/1^; 

*c  +  a 

and  to  find  b  by  the  method  given  in  Note  1,  since  the  same  logarithms 
are  used  in  both  cases. 

Example.    Given  a  =  49,  c  =  50  ;  find  A,  B,  b. 


b  =H1og(c-«)+1°g(c  +  a)] 

c  —  a  =1 

c  +  a         =99 
log  (c  -  a)  =  0.00000 
log  (c  +  a)  =  1.99564 
2)1.99564 
log  b  -  0.99782 

b  =-■  9.95 


log  tan  \B  =  \  [log  (c  —  a) 

+  colog  (c  +  a)] 

log(c-a)     =0.00000 

colog  (c  +  a)  =  8.00436 

2)  8.00436 

log  tan  %B    =9.00218 

%B   =5°  44' 21" 

B    =11°  29' 

A    =78^31' 


20 


TRIGONOMETRY. 


§  16.   Area  of  the  Right  Triangle. 

It  is  shown  in  Geometry  that  the  area  of  a  triangle  is  equal 
to  one-half  the  product  of  the  base  by  the  altitude. 

Therefore,  if  a  and  b  denote  the  legs  of  a  right  triangle,  and 
F  the  area,  F=$ab. 

By  means  of  this  formula  the  area  may  always  be  found 
when  a  and  b  are  given  or  have  been  computed. 
For  example :    Find  the  area,  having  given  : 


Case  I.  (§  10). 
.4  =  34°  28',  c  =  18.75. 
First  find  (as  in  §  10)  log  a 
and  log&. 

]og(F)  —  \oga-\-logb-{-co\og2 
log  a      =    1.02578 
log  b      =    1.18915 
colog2  =    9.69897 

\og(F)  =    1.91390 
F         =  82.016 


Case  IV.  (§  13). 
a  -47.54,  c  =  58.40. 
First  find  (as  in  §  13)  log  a 
and  log  b. 

log  (F) = log  a  +log  b  -f-  colog  2 
log  a      =1.67715 
log  b      =1.53025 
colog  2  =9.69897 

\og(F)  =  2.90637 
F         =  806.06 


Exercise  VI. 

1.  In  Case  II.  give  another  way  of  finding  c,  after  b  has  been 
found. 

2.  In  Case  III.  give  another  way  of  finding  c,  after  a  has 
been  found. 

3.  In  Case  IV.  give  another  way  of  finding  b,  after  the 
angles  have  been  found. 

4.  In  Case  V.  give  another  way  of  finding  c,  after  the  angles 
have  been  found. 

5.  Given  B  and  c  ;  find  A,  a,  b. 

6.  Given  B  and  b  ;  find  A,  a,  c. 

7.  Given  B  and  a;  find  A,  b,  c. 

8.  Given  b  and  c ;  find  A,  B,a. 


THE   RIGHT   TRIANGLE. 


21 


Solve  the  following  triangles : 

9 

Given  : 

Required  : 

a  =  6, 

c  =  12. 

A  =  30°,       B  =  60°, 

6  =  10.392. 

10 

A  -  60°, 

6  =  4. 

B  =  30°,         c  =  8, 

a  =  6.9282. 

/ 

11 

A  -  30°, 

a  =  3. 

B  =  60°,         c  =  6, 

6  =  5.1961. 

/ 

12 

a  =  4, 

6  =  4. 

A  =  5  =  45°,  c  =  5.6568. 

13 

a  =  2, 

c  -  2.82843. 

^  =  £  =  45°,  6  =  2. 

14 

c  =  627, 

J.  =  23°  30'. 

5  =  66°  30',  a  =  250.02, 

6  =  575.0. 

15 

c  -  2280, 

JL  =  28°  5'. 

B  =  61°  55',  a  =  1073.3, 

6  =  2011.6. 

y 

16 

c  -  72.15, 

A  =  39°  34'. 

£  =  50°  26',  a  =  45.958, 

6  =  55.620. 

, 

17 

c-lf 

A  =  36°. 

B  =  54°,        a  =  0.58779, 

6  =  0.80902. 

18 

c  -  200, 

£  -  21°  47'. 

^  =  68°13',  a  =185.73, 

6  -  74.22. 

!•- 

19 

c  =  93.4, 

B  =76°  25'. 

^1  =  13°  35',  a  =  21.936, 

6  =  90.788. 

20 

a  -  637, 

^1=    4°  35'. 

B  =  85°  25',  6  =  7946, 

c=  7971.5. 

21 

a  =  48.532, 

4  =  36°  44'. 

5  =  53°  16',  6  =  65.031, 

c  =  81.144. 

22 

a  =  0.0008, 

A  =  86°. 

B=   4°,        6  =  0.0000559 

,  c  =  0.000802 

23 

6  =  50.937, 

£  =  43°  48'. 

J.  =  46°  12',  a  =  53.116, 

c  -  73.59. 

24 

6  =  2, 

B=   3°  38'. 

A  =  86°  22',  a  =  31.497, 

c  =  31.560. 

25 

a  =  992, 

£  =  76°  19'. 

A  =  13°  41',  6  =  4074.5. 

c  =  4193.6. 

26 

a  =73, 

£  =  68°  52'. 

A  =  21°  8,      6  =  188.86, 

c  =  202.47. 

27 

a  =  2.189, 

B  =  45°  25'. 

^4  =  44°  35',  6  =  2.2211, 

c  =  3.1185. 

s 

28 

6  =  4, 

A  =  37°  56'. 

5  =  52°  4',    a  =  3.1176, 

c  =  5.0714. 

29 

c  =  8590, 

a  =  4476. 

^  =  31°  24',  £  =  58°  36', 

6  =  7332.8. 

30 

c  =  86.53, 

a  =  71.78. 

4  =  56°  3',    £  =  33°  57'. 

6  =  48.324. 

31 

c  =  9.35, 

a  =  8.49. 

A  =  65°  14',  B  -  24°  46', 

6  =  3.917. 

32 

c  =  2194, 

6  =  1312.7. 

A  =  53°  15',  5  =  36°  45', 

a  =  1758. 

33 

c  -  30.69, 

6  =  18.256. 

A  =  53°  30',  B  =  36°  30', 

a  =  24.67. 

34 

a  =  38.313, 

6  =  19.522. 

A  =  63°,        5  =  27°, 

c  =  43. 

35 

a  =  1.2291, 

6  =  14.950. 

A=   4C  42',  B  =  85°  18', 

c=15. 

36 

a  =  415.38, 

6  =  62.080. 

^1  =  81°  30',  J5=    8°  30', 

c  =  420. 

37 

a  -  13.690, 

6  =  16.926. 

J.  =  38°  58',  B  =  51°  2', 

c  =  21.769. 

38 

c  -  91.92, 

a  =  2.19. 

J.  =  1°21'55",  5  =  88°38'5",  6  -  91.894. 

Compute  the  unknown  parts  and  also  the  area,  having  given 

39.  a  =  5,         5  =  6. 

40.  a  =  0.615,  c  =  70L 

41.  b  =  -ty2,     c=V3. 

42.  a  =  7,       ^  =  18°  14'. 

43.  5=12,     ^4  =  29°  8'. 


44.  c  =  68, 

45.  c  =  27, 

46.  a  =  47, 

47.  5  =  9, 


^4  =  69°  54'. 
^  =  44°  4'. 
.5  =  48°  49'. 
.5  =  34°  44'. 


48.    c  =  K462  J9  =  86°4' 


22  TRIGONOMETRY. 


49.  Find  the  value  of  Fin  terms  of  c  and  A. 

50.  Find  the  value  of  .Fin  terms  of  a  and  A. 

51.  Find  the  value  of  Fin  terms  of  b  and  A. 

52.  Find  the  value  of  Fin  terms  of  a  and  c. 

53.  Given  F=  58,    a  =  10  ;    solve  the  triangle. 

54.  Given  F=  18,     b  =  5  ;      solve  the  triangle. 

55.  Given  F=  12,  ^4  =  29° ;  solve  the  triangle. 

56.  Given  F=  100,  c  =  22;    solve  the  triangle. 

57.  Find  the  angles  of  a  right  triangle  if  the  hypotenuse  is 
equal  to  three  times  one  of  the  legs. 

58.  Find  the  legs  of  a  right  triangle  if  the  hypotenuse  =  6, 
and  one  angle  is  twice  the  other. 

59.  In  a  right  triangle  given  c,  and  A  —  nB ;  find  a  and  b. 

60.  In  a  right  triangle  the  difference  between  the  hypote- 
nuse and  the  greater  leg  is  equal  to  the  difference  between  the 
two  legs ;  find  the  angles. 


The  angle  of  elevation  of  an  object  (or  angle  of  depression, 
if  the  object  is  below  the  level  of  the  observer)  is  the  angle 
which  a  line  from  the  eye  to  the  object  makes  with  a  horizon- 
tal line  in  the  same  vertical  plane. 

61.  At  a  horizontal  distance  of  120  feet  from  the  foot  of  a 
steeple,  the  angle  of  elevation  of  the  top  was  found  to  be  60°  30' ; 
find  the  height  of  the  steeple. 

62.  From  the  top  of  a  rock  that  rises  vertically  326  feet  out 
of  the  water,  the  angle  of  depression  of  a  boat  was  found  to  be 
24°  ;  find  the  distance  of  the  boat  from  the  foot  of  the  rock. 

63.  How  far  is  a  monument,  in  a  level  plain,  from  the  eye, 
if  the  height  of  the  monument  is  200  feet  and  the  angle  of  ele- 
vation of  the  top  3°  30'? 

64.  In  order  to  find  the  breadth  of  a  river  a  distance  AB 
was  measured  along  the  bank,  the  point  A  being  directly  op- 
posite a  tree  C  on  the  other  side.  The  angle  A B C was  also 
measured.  If  AB  =  96  feet,  and  ABC=  21°  W,  find  the 
breadth  of  the  river. 

If  ABC=  45°,  what  would  be  the  breadth  of  the  river? 


THE    BIGHT    TRIANGLE.  23 

65.  Find  the  angle  of  elevation  of  the  sun  when  a  tower 
a  feet  high  casts  a  horizontal  shadow  b  feet  long.  Find  the 
angle  when  a  =  120,  b  =  70. 

66.  How  high  is  a  tree  that  casts  a  horizontal  shadow  b  feet 
in  length  when  the  angle  of  elevation  of  the  sun  is  A°  ?  Find 
the  height  of  the  tree  when  b  =  80,  A  =  50°. 

67.  What  is  the  angle  of  elevation  of  an  inclined  plane  if  it 
rises  1  foot  in  a  horizontal  distance  of  40  feet  ? 

68.  A  ship  is  sailing  due  north-east  with  a  velocity  of  10 
miles  an  hour.  Find  the  rate  at  which  she  is  moving  due 
north,  and  also  due  east. 

69.  In  front  of  a  window  20  feet  high  is  a  flower-bed  6  feet 
wide.  How  long  must  a  ladder  be  to  reach  from  the  edge  of 
the  bed  to  the  window  ? 

70.  A  ladder  40  feet  long  may  be  so  placed  that  it  will  reach 
a  window  33  feet  high  on  one  side  of  the  street,  and  by  turn- 
ing it  over  without  moving  its  foot  it  will  reach  a  window  21 
feet  high  on  the  other  side.     Find  the  breadth  of  the  street. 

71.  From  the  top  of  a  hill  the  angles  of  depression  of  two 
successive  milestones,  on  a  straight  level  road  leading  to  the 
hill,  are  observed  to  be  5°  and  15°.  Find  the  height  of  the 
hill. 

72.  A  fort  stands  on  a  horizontal  plain.  The  angle  of  ele- 
vation at  a  certain  point  on  the  plain  is  30°,  and  at  a  point  100 
feet  nearer  the  fort  it  is  45°.     How  high  is  the  fort? 

73.  From  a  certain  point  on  the  ground  the  angles  of  eleva- 
tion of  the  belfry  of  a  church  and  of  the  top  of  the  steeple  were 
found  to  be  40°  and  51°  respectively.  From  a  point  300  feet 
farther  off,  on  a  horizontal  line,  the  angle  of  elevation  of  the 
top  of  the  steeple  is  found  to  be  33°  45'.  Find  the  distance 
from  the  belfry  to  the  top  of  the  steeple. 

74.  The  angle  of  elevation  of  the  top  of  an  inaccessible  fort 
C,  observed  from  a  point  A,  is  12°.  At  a  point  B,  219  feet 
from  A  and  on  a  line  AB  perpendicular  to  AC,  the  angle  ABC 
is  61*  45'.     Find  the  height  of  the  fort. 


24 


TRIGONOMETRY. 


§  17.   The  Isosceles  Triangle. 

An  isosceles  triangle  is  divided  by  the  perpendicular  from 
the  vertex  to  the  base  into  two  equal  right  triangles. 

Therefore,  an  isosceles  triangle  is  determined  by  any  two 
parts  that  determine  one  of  these  right  triangles. 

Let  the  parts  of  an  isosceles  triangle  ABC  (Fig.  13),  among 
which  the  altitude  CD  is  to  be  in- 
cluded, be  denoted  as  follows : 


a  =  one  of  the  equal  sides. 
c  =  the  base. 
h  =3  the  altitude. 
A  =  one  of  the  equal  angles. 
C=  the  angle  at  the  vertex. 


For  example :    Given  a  and  c 
quired  A,  C,  h. 


re- 


1  a       \c        c 

1.  cos^l  =  —  =  - — 

a      2a 

2.  (7+ 2.4  =  180°; 


C=  180°  -2 A  =  2(90°-^). 
3.  h  may  be  found  directly  in  terms  of  a  and  c  from  the 


equation 


#+!=«*, 


which  gives  h  =  V(a  —  he)  (a -f-  £ c). 

But  it  is  better  to  find  the  angles  first,  and  then  find  h  from 
either  one  of  the  two  equations, 


whence 


or 


=  sin^4. 

=  asin^,  or 


h   —^ctdiJiA. 


The  numerical  values  of  A,  C,  and  h  may  be  computed  by 
the  aid  of  logarithms,  as  in  the  case  of  the  right  triangle. 

The  area  F  of  the  triangle  may  be  found,  when  c  and  h  are 
given  or  have  been  found,  by  means  of  the  formula 
F=hch. 


THE   ISOSCELES    TRIANGLE. 


Exercise  VII. 

In  an  isosceles  triangle  : 

1.  Given  a  and  A;  find  C,  c,  h. 

2.  Given  a  and  C;  find  A,  c,  h. 

3.  Given  c  and  A ;  find  C,  a,  h. 

4.  Given  c  and  C;  find  ^4,  a,  h. 
.  5.    Given  h  and  J. ;  find  C,  a,  c. 

6.  Given  h  and  C;  find  J.,  a,  c. 

7.  Given  a  and  h;  find  .4,   (7,  c. 

8.  Given  <?  and  h ;  find  ^4,  (7,  a. 

9.  Given  a  =14.3,       c  =  11 ; 

10.  Given  a  =  0.295,    ^4  =  68°  10' ; 

11.  Given  c  =  2.352,     C=69°49'; 

12.  Given  h  =  7.4847,  ^4  =  76°  14' ; 

13.  Given  a  =  6.71,       A  =  6.60; 

14.  Given  c  =  9,  A  =  20; 

15.  Given  c  =  147,      i^=  2572.5; 

16.  Given  A  =  16.8,     F=  43.68; 

17.  Find  the  value  of  i^in  terms  of  a  and  c. 

18.  Find  the  value  of  -Fin  terms  of  a  and  C. 

19.  Find  the  value  of  JPin  terms  of  a  and  A. 

20.  Find  the  value  of  F  in  terms  of  h  and  C. 

21.  A  barn  is  40  X  80  feet,  the  pitch  of  the  roof  is  45° ;  find 
the  length  of  the  rafters  and  the  area  of  both  sides  of  the  roof. 

22.  In  a  unit  circle  what  is  the  length  of  the  chord  corre- 
sponding to  the  angle  45°  at  the  centre? 

-   23.    If  the  radius  of  a  circle  =  30,  and  the  length  of  a  chord 
«=  44,  find  the  angle  at  the  centre. 

24.  Find  the  radius  of  a  circle  if  a  chord  whose  length  is  5 
subtends  at  the  centre  an  angle  of  133°. 

25.  What  is  the  angle  at  the  centre  of  a  circle  if  the  corre- 
sponding chord  is  equal  to  I  of  the  radius  ? 

26.  Find  the  area  of  a  circular  sector  if  the  radius  of  the 
circle  =  12,  and  the  angle  of  the  sector  =  30c 


find  .4 

,C,h. 

find  c, 

h,F. 

find  a, 

h,F 

find  a, 

c,F. 

find  A, 

C,c. 

find  A, 

,  C,  a. 

find  A, 

,  C,  a,  h. 

find  A, 

C,  a,  c. 

\o 


26 


TRIGONOMETRY. 


§  18.   The  Regular  Polygon. 

Lines  drawn  from  the  centre  of  a  regular  polygon  (Fig.  14) 
to  the  vertices  are  radii  of  the  circumscribed  circle  ;  and  lines 
drawn  from  the  centre  to  the  middle  points  of  the  sides  are 
radii  of  the  inscribed  circle.  These  lines  divide  the  polygon 
into  equal  right  triangles.  Therefore,  a  regular  polygon  is 
determined  by  a  right  triangle  whose  sides  are  the  radius  of 
the  circumscribed  circle,  the  radius  of  the  inscribed  circle,  and 
half  of  one  side  of  the  polygon. 

If  the  polygon  has  n  sides,  the  angle  of  this  right  triangle  at 
the  centre  is  equal  to 

1  /360°\     Qr   180° 
2\  n  J  n  ■ 

If,  also,  a  side  of  the  polygon,  or  one  of  the  above-mentioned 
radii,  is  given,  this  triangle  may  be  solved,  and  the  solution 
gives  the  unknown  parts  of  the  polygon. 

Let, 
n  =  number  of  sides. 
c  =  length  of  one  side. 
r  =  radius  of  circumscribed  circle. 
h  =  radius  of  inscribed  circle. 
p  =  the  perimeter. 
F=  the  area. 

Then,  by  Geometry, 

F=  *  hp. 

r 

Exercise  VIII. 


Fig.  14. 


1. 

Given  n  =  10, 

0-1; 

find  r,  h,  F. 

2. 

Given  n  =  12, 

p=70; 

find  r,  h,  F 

3. 

Given  n  — 18, 

r  =  l; 

find  h,  p,  F. 

4. 

Given  n  =  20, 

r  =  20 

find  h,  c,  F. 

5. 

Given  n  —  8, 

h  =  l; 

find  r,  c,  F. 

6. 

Given  w  =  ll, 

F=20; 

find  r,  h,  c. 

7. 

Given  n  =  7, 

F=7; 

find  r,  h,  p. 

THE    REGULAR    POLYGON.  27 

8.  Find  the  side  of  a  regular  decagon  inscribed  in  a  unit 
circle. 

9.  Find  the  side  of  a  regular  decagon  circumscribed  about 
a  unit  circle. 

10.  If  the  side  of  an  inscribed  regular  hexagon  is  equal  to  1, 
find  the  side  of  an  inscribed  regular  dodecagon. 

11.  Given  n  and  c,  and  let  b  denote  the  side  of  the  inscribed 
regular  polygon  having  2w  sides;  find  b  in  terms  of  n  and  c. 

12.  Compute  the  difference  between  the  areas  of  a  regular 
octagon  and  a  regular  nonagon  if  the  perimeter  of  each  is  16. 

13.  Compute  the  difference  between  the  perimeters  of  a 
regular  pentagon  and  a  regular  hexagon  if  the  area  of  each  is  12. 

14.  From  a  square  whose  side  is  equal  to  1  the  corners  are 
cut  away  so  that  a  regular  octagon  is  left.  Find  the  area  of 
this  octagon. 

15.  Find  the  area  of  a  regular  pentagon  if  its  diagonals  are 
each  equal  to  12. 

16.  The  area  of  an  inscribed  regular  pentagon  is  331.8; 
find  the  area  of  a  regular  polygon  of  11  sides  inscribed  in  the 
same  circle. 

17.  The  perimeter  of  an  equilateral  triangle  is  20;  find 
the  area  of  the  inscribed  circle. 

18.  The  area  of  a  regular  polygon  of  16  sides,  inscribed  in 
a  circle,  is  100 ;  find  the  area  of  a  regular  polygon  of  15  sides, 
inscribed  in  the  same  circle. 

19.  A  regular  dodecagon  is  circumscribed  about  a  circle, 
the  circumference  of  which  is  equal  to  1 ;  find  the  perimeter 
of  the  dodecagon. 

20.  The  area  of  a  regular  polygon  of  25  sides  is  equal  tc 
40;  find  the  area  of  the  ring  comprised  between  the  circum- 
ferences of  the  inscribed  and  the  circumscribed  circles. 


CHAPTER  III. 

GONIOMETRY. 
§  21.   Definition  of  Goniometry. 

In  order  to  prepare  the  way  for  the  solution  of  an  oblique 
triangle,  we  now  proceed  to  extend  the  definitions  of  the 
trigonometric  functions  to  angles  of  all  magnitudes,  and  to 
deduce  certain  useful  relations  of  the  functions  of  different 
angles. 

That  branch  of  Trigonometry  which  treats  of  trigonometric 
functions  in  general,  and  of  their  relations,  is  called  Goniometry. 


§  22.   Angles  of  any  Magnitude. 

Let  the  radius.  OP  of  a  circle  (Fig.  16)  generate  an  angle  by 

turning  about  the  centre  0.  This 
angle  will  be  measured  by  the 
arc  described  by  the  point  P; 
and  it  may  have  any  magnitude, 
because  the  arc  described  by  P 
may  have  any  magnitude. 

Let  the  horizontal  line  OA  be 
the  initial  position  of  OP,  and 
let  OP  revolve  in  the  direction 
shown  by  the  arrow,  or  opposite 
to  the  way  clock-hands  revolve. 
Let,  also,  the  four  quadrants  into 
which  the  circle  is  divided  by  the  horizontal  and  vertical 
diameters  AA\  BB\  be  numbered  I.,  II,  III.,  IV.,  in  the 
direction  of  the  motion. 


GONIOMETRY.  29 


During  one  revolution  OP  will  form  with  OA  all  angles  from 
0°  to  360°.  Any  particular  angle  is  said  to  be  an  angle  of  the 
quadrant  in  which  OP  lies ;  so  that, 

Angles  between      0°  and    90°  are  angles  of  Quadrant  I. 

Angles  between    90°  and  180°  are  angles  of  Quadrant  II. 

Angles  between  180°  and  270°  are  angles  of  Quadrant  III. 

Angles  between  270°  and  360°  are  angles  of  Quadrant  IV. 

If  OP  make  another  revolution,  it  will  describe  all  angles 
from  360°  to  720°,  and  so  on. 

If  OP,  instead  of  making  another  revolution  in  the  direction 
of  the  arrow,  be  supposed  to  revolve  backwards  about  O,  this 
backward  motion  tends  to  undo  or  cancel  the  original  forward 
motion.  Hence,  the  angle  thus  generated  must  be  regarded 
as  a  negative  angle ;  and  this  negative  angle  may  obviously 
have  any  magnitude.  Thus  we  arrive  at  the  conception  of  an 
angle  of  any  magnitude,  positive  or  negative. 

§  23.   General  Definitions  of  the  Functions. 

The  definitions  of  the  trigonometric  functions  may  be  ex- 
tended to  all  angles,  by  making  the  functions  of  any  angle 
equal  to  the  line  values  in  a  unit  circle  drawn  for  the  angle 
in  question,  as  explained  in  §  3.  Put  the  lines  that  represent 
the  sine,  cosine,  tangent,  and  cotangent  must  be  regarded  as 
negative,  if  they  are  opposite  in  direction  to  the  lines  that  repre- 
sent the  corresponding  functions  of  an  angle  in  the  first  quad- 
rant ;  and  the  lines  that  represent  the  secant  and  cosecant  must 
be  regarded  as  negative,  if  they  are  opposite  in  direction  to  the 
moving  radius. 

Figs.  17-20  show  the  functions  drawn  for  an  angle  A  OP  in 
each  quadrant  taken  in  order.  In  constructing  them,  it  must 
be  remembered  that  the  tangents  to  the  circle  are  always 
drawn  through  A  and  P,  never  through  A1  or  P'. 

Let  the  angle  A  OP  be  denoted  by  x;  then,  in  each  figure 
the  absolute  values  of  the  functions,  that  is,  their  values  with- 
out  regard  to  the  signs  -f-  and  — ,  are  as  follows  : 


30 


TRIGONOMETRY. 


sin  x  =  MP, 

cos  a;  =  OJf, 

5 


tan.r 
cot  a; 


Keeping  in  mind  the  position  of  the  points  A  and  JB,  we  may 
define  in  words  the  first  four  functions  of  the  angle  x  thus : 


cos  x 


the  vertical  projection  of  the  moving  radius  ; 
the  horizontal  projection  of  the  moving  radius ; 
the  distance  measured  along  a  tangent  to  the  circle 
from  the  beginning  of  the  first  quadrant  to  the 
moving  radius  produced ; 
the  distance  measured  along  a  tangent  to  the  circle 
from  the  end  of  the  first  quadrant  to  the  moving 
radius  produced. 
Sec  x  and   esc  x  are  the  distances  from  the   centre   of  the 
circle  measured  along  the  moving  radius  produced  to  the  tan- 
gent and  cotangent  respectively. 


GONIOMETRY. 


31 


§  24.   Algebraic  Signs  of  the  Functions. 

The  lengths  of  the  lines,  defined  above  as  the  functions  of 
any  angle,  are  expressed  numerically  in  terms  of  the  radius 
of  the  circle  as  the  unit.  But,  before  these  lengths  can  be 
treated  as  algebraic  quantities,  they  must  have  the  sign  -f  or 
—  prefixed,  according  to  the  condition  stated  in  §  23. 

The  reason  for  this  condition  lies  in  that  fundamental  rela- 
tion between  algebraic  and  geometric  magnitudes,  in  virtue  of 
which  contrary  signs  in  Algebra  correspond  to  opposite  direc- 
tions in  Geometry. 

The  sine  MP  and  the  tangent  A  T  always  extend  from  the 
horizontal  diameter,  but  sometimes  upwards  and  sometimes 
downwards ;  the  cosine  OM  and  the  cotangent  B&  always 
extend  from  the  vertical  diameter,  but  sometimes  towards  the 
right  and  sometimes  toivards  the  left.  The  functions  of  an  angle 
in  the  first  quadrant  are  assumed  to  be  positive.     Therefore, 

1.  Sines  and  tangents  extending  from  the  horizontal  diam- 
eter upwards,  are  positive  ;  downwards,  negative. 

2.  Cosines  and  cotangents  extending  from  the  vertical  diame- 
ter towards  the  right,  are  positive;  towards  the  left,  are  nega- 
tive. 

The  signs  of  the  secant  and  cosecant  are  always  made  to 
agree  with  those  of  the  cosine  and  sine  respectively.  This 
agreement  is  secured  if  secants  and  cosecants  extending  from 
the  centre,  in  the  direction  of  the  moving  radius,  are  consid- 
ered positive  ;  in  the  opposite  direction,  negative. 

Hence,  the  signs  of  the  functions  for  each  quadrant  are : 


Sine  and  cosecant 

I. 
+ 

II. 

III. 

IV. 

+ 

— 

— 

Cosine  and  secant 

+ 

— 

— 

+ 

Tangent  and  cotangent .... 

+ 

— 

+ 

— 

32 


TRIGONOMETRY. 


§  25.    Functions  of  a  Variable  Angle. 

Let  the  angle  z  increase  continuously  from  0°  to  360° ; 
what  changes  will  the  values  of  its  functions  undergo  ? 

It  is  easy,  by  reference  to  Figs.  21-24,  to  trace  these 
changes  throughout  all  the  quadrants. 


8 

B 

A'\               i  \ 

^  J 

KO 

1 

rig.  -4. 


1.  The  Sine.  In  the  first  quadrant,  the  sine  MP  increases 
from  0  to  1 ;  in  the  second,  it  remains  positive,  and  decreases 
from  1  to  0 ;  in  the  third,  it  is  negative,  and  increases  in  abso- 
lute value  from  0  to  1 ;  in  the  fourth,  it  is  negative,  and 
decreases  in  absolute  value  from  1  to  0. 


GONIOMETRY.  33 


2.  The  Cosine.  In  the  first  quadrant,  the  cosine  OM  de- 
creases from  1  to  0;  in  the  second,  it  becomes  negative  and 
increases  in  absolute  value  from  0  to  1;  in  the  third,  it  is 
negative  and  decreases  in  absolute  value  from  1  to  0;  in  the 
fourth,  it  is  positive  and  increases  from  0  to  1. 

3.  The  Tangent.  In  the  first  quadrant,  the  tangent  AT 
increases  from  0  to  oo ;  in  the  second  quadrant,  as  soon  as 
the  angle  exceeds  90°  by  the  smallest  conceivable  amount,  the 
moving  radius  OP,  prolonged  in  the  direction  opposite  to  that 
of  OP,  will  cut  A  T  at  a  point  T  situated  very  far  below  A  ; 
hence,  the  tangents  of  angles  near  90°  in  the  second  quad- 
rant have  very  large  negative  values.  As  the  angle  increases, 
the  tangent  -4  jP  continues  negative  but  diminishes  in  absolute 
value.  When  x  =  180°,  then  I7  coincides  with  A,  and  tan  180° 
=  0.  In  the  third  quadrant,  the  tangent  is  positive  and  in- 
creases from  0  to  oo ;  in  the  fourth,  it  is  negative  and  decreases 
in  absolute  value  from  oo  to  0. 

4.  The  Cotangent.  In  the  first  quadrant,  the  cotangent  BS 
decreases  from  oo  to  0 ;  in  the  second  quadrant,  it  is  negative 
and  increases  in  absolute  value  from  0  to  oo  ;  in  the  third  and 
fourth  quadrants,  it  has  the  same  sign,  and  undergoes  the  same 
changes  as  in  the  first  and  second  quadrants  respectively. 

5.  The  Secant.  In  the  first  quadrant,  the  secant  OT  in- 
creases from  1  to  oo ;  in  the  second  quadrant,  it  becomes 
negative  (being  measured  in  the  direction  opposite  to  that  of 
OP),  and  decreases  in  absolute  value  from  oo  to  1,  so  that 
sec  180°  =  — 1;  in  the  third  quadrant,  it  continues  negative, 
and  increases  in  absolute  value  from  1  to  oo ;  in  the  fourth 
quadrant,  it  is  positive,  and  decreases  from  oo  to  1. 

6.  The  Cosecant.  In  the  first  quadrant,  the  cosecant  OS 
decreases  from  oo  to  1 ;  in  the  second  quadrant,  it  remains 
positive,  and  increases  from  1  to  oo ;  in  the  third  quadrant,  it 
becomes  negative,  and  decreases  in  absolute  value  from  oo  to 
1,  so  that  esc  270°  =  — 1 ;  in  the  fourth  quadrant,  it  is  nega- 
tive, and  increases  in  absolute  value  from  1  to  oo. 


34 


TRIGONOMETRY, 


The  limiting  values  of  the  functions  are  as  follows  : 


Sine 

Cosine   ....... 

Tangent  

Cotangent  

Secant  

Cosecant 

0° 

90° 

180° 

270° 

360* 

±0 
1 

±0 

=b  00 

1 
db  00 

1 

±  0 

rb  00 

±0 

rb  00 
1 

±  0 

-1 

=fc  0 
db  oo 
-1 

=b  00 

-  1 

±0 

=b  00 

±0 

zh  00 

-  1 

±  0 

1 

±0 

rb  00 

1 

rb  00 

Sines  and  cosines  extend  from  + 1  to  —  1  ;  tangents  and  co- 
tangents from  +00  to  —  oo  ;  secants  and  cosecants  from  -f  oo 
to  + 1,  and  from  —  1  to  —  oo. 

In  the  table  given  above  the  double  sign  ±  is  placed  before  0  and 
oo.  From  the  preceding  investigation  it  appears  that  the  functions  always 
change  sign  in  passing  through  0  and  oo ;  and  the  sign  +  or  —  prefixed 
to  0  or  oo  simply  shows  the  direction  from  which  the  value  is  reached. 

Take,  for  example,  tan  90° :  The  nearer  an  acute  angle  is  to  90°,  the 
greater  the  positive  value  of  its  tangent ;  and  the  nearer  an  obtuse  angle 
is  to  90°,  the  greater  the  negative  value  of  its  tangent.  When  the  angle 
is  90°,  OP  (Fig.  21)  is  parallel  to  AT,  and  cannot  meet  it.  But  tan  90° 
may  be  regarded  as  extending  either  in  the  positive  or  in  the  negative 
direction ;  and  according  to  the  view  taken,  it  will  be  +  oo  or  —  o®. 


§  26.    Functions  of  Angles  Larger  than  360°. 

It  is  obvious  that  the  functions  of  360°  -f.r  are  the  same 
both  in  sign  and  in  absolute  value  as  those  of  x ;  for  the  mov- 
ing radius  has  the  same  position  in  both  cases.  In  general,  if 
n  denote  any  positive  whole  number, 

The  functions  of (nX  360°  +  x)  are  the  same  as  those  of  x. 

For  example :  the  functions  of  2200°  =  the  functions  of 
(6  x  360°  +  40°)  =  the  functions  of  40°. 


GONIOMETRY.  35 


§  27.   Extension  of  Formulas  [l]-[3]  to  all  Angles. 

The  Formulas  established  for  acute  angles  in  §  5  hold  true 
for  all  angles.     Thus,  Formula  [1], 

sin2o;  -f-  cos2a;  =  1, 
is  universally  true  ;  for,  whether  MP  and  OM  (Figs.  21-24) 
are  positive  or  negative,  MP^  and  OM"  are  always  positive, 
and  in  each  quadrant  MP2  -f-  OM2  =  OP2  =  1. 

Also,  Formulas 

ro1     .  sin  x 

z       tan#  = , 

L  cos  x 

C  sin  xX  csc#  =  1, 
and  [3]  <  cos  x  X  sec  x  =  1, 

(.  tana;X  cot#=l, 
are  universally  true ;  for  the  algebraic  signs  of  the  functions, 
as  given  in  the  table  at  the  end  of  §  24,  agree  with  those  in 
Formulas  [2]  and  [3] ;  and  with  regard  to  the  absolute  values, 
we  have  in  each  quadrant  from  the  similar  triangles  OMP, 
OAT,  0B8,  (Figs.  21-24)  the  proportions 

AT  :OA=MP:OM, 

MP:0P=  OB  :  OS, 

OM  :  OP=OA   :0T, 

AT  :OA  =  OB  :BS, 
which,  by  substituting  1  for  the  radius,  and  the  right  names 
for  the  other  lines,  are  easily  reduced  to  the  above  formulas. 
Formulas  [l]-[3]  enable  us,  from  a  given  value  of  one  func- 
tion, to  find  the  absolute  values  of  the  other  five  functions,  and 
also  the  sign  of  the  reciprocal  function.  But  in  order  to  deter- 
mine the  proper  signs  to  be  placed  before  the  other  four 
functions,  we  must  know  the  quadrant  to  which  the  angle  in 
question  belongs ;  or  what  amounts  to  the  same  thing,  the  sign 
of  any  one  of  these  four  functions ;  for,  by  reference  to  the 
Table  of  Signs  (§  24)  it  will  be  seen  that  the  signs  of  any  two 
functions  that  are  not  reciprocals  determine  the  quadrant  to 
which  the  angle  belongs. 


36  TRIGONOMETRY 


Example.  Given  sin#  =  -f--fJ  and  tana;  negative;  find  the 
values  of  the  other  functions. 

Since  sin  x  is  positive,  x  must  be  an  angle  in  Quadrant  I.  or 
in  Quadrant  II.;  but,  since  also  tana;  is  negative,  Quadrant  I. 
is  inadmissable. 

By  [1],  cos*  =  ±Vl-if  =  ±  f  • 

Since  the  angle  is  in  Quadrant  II.  the  minus  sign  must  be 
taken,  and  we  have 

cos#  =  —  f. 
By  [2]  and  [3], 

tan#  =  —  -|,     cot#  =  —  f,     sec#  =  —  -§•,     csca;  =  J. 


Exercise  IX. 

1.  Construct  the  functions  of  an  angle  in  Quadrant  II. 
What  are  their  signs  ? 

2.  Construct  the  functions  of  an  angle  in  Quadrant  III. 
What  are  their  signs  ? 

3.  Construct  the  functions  of  an  angle  in  Quadrant  IV. 
What  are  their  signs  ? 

4.  What  are  the  signs  of  the  functions  of  the  following 
angles:  340°,  239°,  145°,  400°,  700°,  1200°,  3800°? 

5.  How  many  angles  less  than  360°  have  the  value  of  the 
sine  equal  to  +-?->  and  in  what  quadrants  do  they  lie? 

6.  How  many  values  less  than  720°  can  the  angle  x  have 
if  cos  X  s=  4"  f ,  an(i  m  what  quadrants  do  they  lie  ? 

7.  If  we  take  into  account  only  angles  less  than  180°,  how 
many  values  can  x  have  if  sin  a;  =  %  ?  if  cos  x  =  -J-  ?  if  cos  a;  = 
-f?  if  tan£  =  f?  if  cot:r  =  -7? 

8.  Within  what  limits  must  the  angle  x  lie  if  cos  x  =  —  |  ? 
if  cot  x  =  4  ?  if  sec  x  =  80  ?  if  esc  x  =  —  3  ?  (z  to  be  less  than 
360°.) 

9.  In  what  quadrant  does  an  angle  lie  if  sine  and  cosine 
are  both  negative  ?  if  cosine  and  tangent  are  both  negative  ? 
if  the  cotangent  is  positive  and  the  sine  negative  ? 


GONIOMETRY.  37 


10.  Between  0°  and  3600°  how  many  angles  are  there  whose 
sines  have  the  absolute  value  f  ?  Of  these  sines  how  many 
are  positive  and  how  many  negative  ? 

11.  In  finding  cos#  by  means  of  the  equation  cos#  = 
—  Vl  —  sin2o:,  when  must  we  choose  the  positive  sign  and  when 
the  negative  sign  ? 

12.  Given  cos  x  =  —  V^  ;  find  the  other  functions  when  x  is 
an  angle  in  Quadrant  II. 

13.  Given  tan#=  V3 ;  find  the  other  functions  when  x  is 
an  angle  in  Quadrant  III. 

14.  Given  sec  x  =  +  7,  and  tan  x  negative  ;  find  the  other 
functions  of  x. 

15.  Given  cot  x  =  —  3  ;  find  all  the  possible  values  of  the 
other  functions. 

16.  What  functions  of  an  angle  of  a  triangle  may  be  nega- 
tive ?     In  what  case  are  they  negative  ? 

17.  What  functions  of  an  angle  of  a  triangle  determine  the 
angle,  and  what  functions  fail  to  do  so  ? 

18.  Why  may  cot  360°  be  considered  equal  either  to  -f  oo 
or  to  —  oo  ? 

19.  Obtain  by  means  of  Formulas  [l]-[3]  the  other  func- 
tions of  the  angles  given  : 

(i.)  tan  90°  =  oo.  (iii.)  cot  270°  =  0. 

(ii.)  cos  180°  =  —  1.  (iv.)  esc  360°  =— oo. 

20.  Find  the  values  of  sin  450°,  tan  540°,  cos  G30°,  cot  720°, 
sin  810°,  esc  900°. 

21.  For  what  an^le  in  each  quadrant  are  the  absolute  values 
of  the  sine  and  cosine  alike  ?   tf$  *  -  / *>$  -  $  %  $ .  -3  /  f 

Compute  the  values  of  the  following  expressions : 
^22.    a  sin  0°  +  b  cos  90°  -c  tan  180°.    0 

23.  acos90o-Z>tanl80°  +  ccot90°.  O 

24.  a  sin  90°  -  b  cos  360°  +  (a  -  b)  cos  180°.  O 

25.  (a2  -  &2)  cos  360°- 4  a&  sin  270°.    «' 


4. 


38 


TRIGONOMETRY. 


< 

N 

/        M 

P 

SV 

H 

8 

B' 

Fig.  SI 


28.   Reduction  of  Functions  to  the  First  Quadrant. 

In  a  unit  circle  (Fig.  25)  draw  two  diameters  PR  and  QS 

equally  inclined  to  the  horizon- 
tal diameter  AA',  or  so  that  the 
angles  AOP,  A'OQ,  A'OR,  and 
-40$  shall  be  equal.  From  the 
points  P,  Q,  P,  S  let  fall  per- 
pendiculars to  AA1 ;  the  four 
right  triangles  thus  formed,  with 
a  common  vertex  at  0,  are  equal ; 
because  they  have  equal  hypote- 
nuses (radii  of  the  circle)  and 
equal  acute  angles  at  0.  There- 
fore, the  perpendiculars  PM, 
QJSf,  RN,  SM,  are  equal.  Now  these  four  lines  are  the  sines 
of  the  angles  A  OP,  AOQ,  AOP,  and  AOS,  respectively. 
Therefore,  in  absolute  value, 

sin  AOP  =  sin  AOQ  =  sin  AOR  =  sin  AOS. 

And  from  §  27  it  follows  that  in  absolute  value  the  cosines 
of  these  angles  are  also  equal ;  and  likewise  the  tangents,  the 
cotangents,  the  secants,  and  the  cosecants.* 

Hence,  for  every  acute  angle  (AOP)  there  is  an  angle  in  each 
of  the  higher  quadrants  whose  functions,  in  absolute  value,  are 
equal  to  those  of  this  acute  angle. 

Let  A  AOP  =x,  /.POB^y;  then  #-fy  =  90°,  and  the 
functions  of  x  are  equal  to  the  co-named  functions  of  y  (§  4) ; 
further, 

A  AOQ  (in  Quadrant  II.)   -  180°  -  x  =    90°  +  y, 
ZAOR  (in  Quadrant  III.)  =  180°  +  x  =  270°  -  y, 
Z  AOS  (in  Quadrant  IV.)  =  360°  -  x  =  270°  +  y. 
Hence,  if  we  prefix  in  each  case  the  proper  sign  (§  24),  we 
have  the  two  following  series  of  Formulas : 


*  In  future,  secants  and  cosecants  will  be  disregarded.     They  may  be  found 
by  [3]  if  wanted,  but  are  seldom  used  in  computations. 


GONIOMETRY.  39 


Angle  in  Quadrant  II 

sin  (180°  —  x)  =      sin  x.  sin  (90°  -J-  y)  =      cos  y. 

cos  (180°  —  x)  =  —  cos  a?.  cos  (90°  +y)  =  —  sin  y. 

tan  (180°  -  a;)  =  -  tan  x.  tan  (90°  +  y)  =  -  cot  y. 

cot  (180°  —  x)  =  —  cot  a:.  cot  (90°  +  y)  =  —  tany. 

Angle  in  Quadrant  III 

sin  (180°  +  a?)  =  —  sin  x.  sin  (270°  ~y)==  —  cos  y. 

cos  (180°  -j-  a?)  =  —  cos  $.  cos  (270°  -  -  y)  =  —  sin  y. 

tan  (180° +  a;)=      tanar.  tan  (270° -- y)  =      coty. 

cot  (180°  +  x)  =      cot  x.  cot  (270°  -  y)  =      tan  y. 

Angle  in  Quadrant  IV. 

sin  (360°  —  x)  —  —  sin  x.  sin  (270°  +  y)  ==  —  cos  y. 

COB  (360°  —  x)  =      cos  07.  cos  (270°  +  y)  ==      sin  y. 

tan  (360°  -  x)  =  -  tan  a:.  tan  (270°  +  y)  =  -  cot  y . 

cot  (360°  -  a:)  =  -  cot  x.  cot  (270°  +  y)  =  -  tany. 

Hence,  by  selecting  the  right  formulas, 

The  functions  of  all  angles  can  be  reduced  to  the  functions  of 
angles  not  greater  than  45°.  Thus,  to  find  the  functions  of 
220°  and  230°,  we  should  consider  220°  as  (180° +  40°),  but 
230°  as  (270°  -  40°). 

It  is  evident  from  these  formulas  that, 

If  an  acute  angle  be  added  to  or  subtracted  from  180°  or  360°, 
the  functions  of  the  resulting  angle  are  equal  in  absolute  value 
to  the  like-named  functions  of  the  acute  angle;  but  if  an  acute 
angle  be  added  to  or  subtracted  from  90°  or  270°,  the  functions 
of  the  resulting  angle  are  equal  in  absolute  value  to  the  co-named 
functions  of  the  acute  angle. 

It  is  evident  from  the  formulas  for  (180°  —  x)  that, 

A  given  value  of  a  sine  determines  two  supplementary  angles, 
one  acute,  the  other  obtuse ;  a  given  value  of  any  other  function 
{except  the  cosecant)  determines  only  one  angle :  acute  if  the 
value  is  positive,  obtuse  if  the  value  is  negative. 


40 


TRIGONOMETRY. 


§  29.   Angles  whose  Difference  is  90°. 

The  general  form  of  two  such  angles  is  x  and  90°  +  x}  and 
they  must  lie  in  adjoining  quadrants.     The  relations  between 

their  functions  were  found  in 
§  28,  but  only  for  the  case  when 
x  is  acute.  These  relations,  how- 
ever, may  be  shown  to  hold  true 
for  all  values  of  x. 

In  a  unit  circle  (Fig.  26)  draw 
two  diameters  PB  and  QS  per- 
pendicular to  each  other,  and 
let  fall  to  A  A'  the  perpendicu- 
lars PM,  QH,  BK,  and  SJV. 
The  right  triangles  OMP,  O&Q, 
OKB,  and  ONS  are  equal,  because  they  have  equal  hypote- 
nuses and  equal  acute  angles  POM,  OQH,  BOK,  and  OSN. 

Therefore,  OM  =  QH=  OK  =  iV#, 

and  PM=  OH=  KB  =  ON. 

Hence,  taking  also  into  account  the  algebraic  sign, 

sin  AOQ=      cos  AOP;  sin  A08  =      cos  AOB; 

cos  AOQ  =  —  sin  AOP;  cos  AOS  =  —  sin  AOB; 

sin  AOB  =     cos  AOQ;  sin  (360°  +  A  OP)  =     cos  AOS; 

cos  AOB  =  -  sin  AOQ ;  cos  (360°  +  AOP)  =  -  sin  AOS. 

In  all  these  equations,  if  x  denote  the  angle  on  the  right-hand 
side,  the  angle  on  the  left-hand  side  will  be  90°  -\-x.  There- 
fore, if  x  be  an  angle  in  any  one  of  the  four  quadrants, 

sin  (90°  +  x)  =      cos  x, 

cos  (90°  -\-x)  =  —  sin  x. 

And,  by  §  27,       tan  (90°  +  x)  =  -  cot  x, 

cot  (90°  +  x)  =  —  tan  x. 

In  like  manner,  it  can  be  shown  that  all  the  formulas  of 
§  28  hold  true,  whatever  be  the  value  of  the  angle  x. 


GONIOMETRY.  41 


§  30.   Functions  of  a  Negative  Angle. 

If  the  angle  A  OP  (Fig.  25)  is  denoted  by  x,  the  equal  angle 
AOS,  generated  by  a  backward  rotation  of  the  moving  radius 
from  the  initial  position  OA,  will  be  denoted  by  —  x.  It  is 
obvious  that  the  position  OS  of  the  moving  radius  for  this 
tngle  is  identical  with  its  position  for  the  angle  360°  —  x. 
Therefore,  the  functions  of  the  angle  —  x  are  the  same  as  those 
of  the  angle  360°  -  x ;  or  (§  28), 

sin  (-—  x)  =  —  sin  x,  tan  (—  x)  =  —  tan  x, 

cos  (—  x)  =      cos  x,  cot  (—  x)  =  —  cot  x. 


Exercise  X. 

1.  Express  sin  250°  in*  terms  of  the  functions  of  an  acute 
angle  greater  than  45°,  and  also  in  terms  of  the  functions  of 
an  acute  angle  less  than  45°. 

Ans.    1.  sin 250°  =  sin (180° +  70°)  = -sin 70°. 
2.  sin  250°  =  sin  (270°  -  20°)  =  -  cos  20°. 

Express  the  following  functions  in  terms  of  the  functions  of 
angles  less  than  45°  : 

sin  204°. 
cos  359°. 
tan  300°. 
cot  264°. 
sec  244°. 
esc  271°. 

Express  all  the  functions  of  the  following  negative  angles  in 
terms  of  those  of  positive  angles  less  than  45°  : 

20.  -75°.  22.   -200°.  24.   -52°37f. 

21.  -127°.  23.   -345°.  25.   -196°  54'. 
26.    Find  the  functions  of  120°. 

Hint.   120°  -  180°  -  60°,  or,  120°  =  90°  +  30°  ;  then  apply  \  28. 


2. 

sin  172°. 

8. 

3. 

cos  100°. 

9. 

4. 

tan  125°. 

10. 

5. 

cot   91°. 

11. 

6. 

sec  110°. 

12. 

7. 

esc  157°. 

13. 

14. 

sin  163°  49'. 

15. 

cos  195°  33'. 

16. 

tan269°  15f. 

17. 

cot  139°  17'. 

18. 

sec  299°  45'. 

19. 

esc   92°  25'. 

42  TRIGONOMETRY. 


Find  the  functions  of  the  following  angles : 

27.  135°.  29.   210°.  31.   240°.  33.   -30°. 

28.  150°.  30.   225°.  32.   300°.  34.   -225°. 

35.  Given  sin  x  =  —  VJ,  and  cos#  negative  ;  find  the  other 
functions  of  x,  and  the  value  of  x. 

36.  Given  cot#  =  —  V3,  and  x  in  Quadrant  II.;  find  the 
other  functions  of  x,  and  the  value  of  x. 

37.  Find  the  functions  of  3540°. 

38.  What  angles  less  than  360°  have  a  sine  equal  to  —J? 
a  tangent  equal  to  —  V3? 

39.  Which  of  the  angles  mentioned  in  Examples  27-34  have 
a  cosine  equal  to  —  VT?  a  cotangent  equal  to  -~-V3? 

40.  What  values  of  x  between  0°  and  720°  will  satisfy  the 
equation  sin  x  =  +  £  ? 

41.  In  each  of  the  following  cases  find  the  other  angle  be- 
tween 0°  and  360°  for  which  the  corresponding  function  (sign 
included)  has  the  same  value:  sin  12°,  cos 26°,  tan 45°,  cot 72°; 
sin  191°,  cos  120°,  tan  244°,  cot 357°. 

42.  Given  tan  238°  =  1.6;  find  sin  122°. 

43.  Given  cos 333°  =  0.89;  findtanll7°. 
Simplify  the  following  expressions : 

44.  a  cos  (90°  -  x)  +  b  cos  (90°  +  ^). 

45.  m cos (90° -a;)  sin (90°  —  x). 

46.  (a  -  b)  tan  (90°  -  x)  +  (a  +  b)  cot  (90°  +  x). 

47.  a?+b2-2abcos(180°-x). 

48.  sin  (90°  +  x)  sin  (180°  +  x) + cos  (90°  +  x)  cos  (180°  -  x). 

49.  cos(180°+a:)cos(270o-y)~sin(180o+^)sin(270o-<y). 

50.  tan  x  +  tan  (-  y)  —  tan  (180°  —  y). 

51.  For  what  values  of  x  is  the  expression  sin#  +  cos  a; 
positive,  and  for  what  values  negative  ?  Represent  the  result 
by  a  drawing  in  which  the  sectors  corresponding  to  the  nega- 
tive values  are  shaded. 

52.  Answer  the  question  of  last  example  for  sin  x  —  cos x. 

53.  Find  the  functions  of  (x  —  90°)  in  terms  of  the  functions 
of  x. 

54.  Find  the  functions  of  (x  — 180°)  in  terms  of  the  functions 


of  x. 


GONIOMETRY. 


43 


§  31.   Functions  of  the  Sum  of  Two  Angles. 

In  a  unit  circle  (Fig.  27)  let  the  angle  AOB  =  x,  the  angle 
BOC=y\  then  the  angle  AOC= 
x  +  y. 

In  order  to  express  sin(#-f  y) 
and  cos  (x  -f-  y)  in  terms  of  the 
sines  and  cosines  of  x  and  y,  draw 
OF  A.  OA,  CB  _L  OB,  BE±  OA, 
BG±CF;  then  00  =  sin  y,  OB 
=  cosy,  and  the  angle  BCG  = 
the  angle  GDO  =  x.    Also, 

sin  (x  +  y)  =  CF=  DE  +  CG. 

BE 
OB 

CG 
CB 
Therefore,      sin  (x  -f  y)  =  sinx  cos y  +'cos  x  sin y 

Again,  cos  (a;  +  y)  =  OF=  OE-  BG. 

OE 
OB 
BG 


i\x ;     hence,  BE=  sin#  X  OB  =  sin.x'  cosy. 
=  cos  x ;     hence,  CG  =  cos  a;  X  (7J9  =  cos  a;  siny. 


m 


cos:£ 


CB 


hence,  OE-=  cos  x  X  OB  =  cos  a;  cosy, 
hence,  BG  =  sin  a;  X  6yZ>  =  sin  a;  siny. 


Therefore ,      cos(x  -j-  y)  =  cos  x  cos  y  —  sin  x  sin  y. 


In  this  proof  x  and  y,  and  also  the  sum  x-\-y,  are  assumed 

to  be  acute  angles.  If  the  sum 
x-\-y  of  the  acute  angles  x  and 
y  is  obtuse,  as  in  Fig.  28,  the 
proof  remains,  word  for  word, 
the  same  as  above,  the  only  dif- 
ference being  that  the  sign  of 
Oi^will  be  negative,  as  BG  is 

now  greater  than  OE.     The  above  formulas,  therefore,  hold 

true  for  all  acute  angles  x  and  y. 


44  TRIGONOMETRY. 


If  these  formulas  hold  true  for  any  two  acute  angles  x  and 
y,  they  hold  true  when  one  of  the  angles  is  increased  by  90°. 
Thus,  if  for  x  we  write  x'  =  90°  +  a;,  then,  by  §  29, 

sin  (V  -f  y)  ==  sin  (90°  -f-  z  +  y)  =      cos  (a;  +  y), 
cos  (a;'  -j-  y)  =  cos  (90°  +  a;  +  y)  =  —  sin  (a;  -f  y). 

Hence,  by  [5],  sin(V+  y)  =s      cosa;  cosy  —  sina;  siny, 
by  [4],  cos(a;'+  y)  =  —  sina;  cosy  —  cos  a;  siny. 

Now,  by  §  29,  cos  a;  =      sin  (90°  -f  x)  =      sin  x', 
sin  x  =  —  cos  (90°  -f  x)  =  —  cos  a;'. 

Therefore,  by  putting  sina;'  for  cos  a;,  and  —cos  a;'  for  sina;, 
in  the  right-hand  members  of  the  above  equations, 

sin  (xf  +  y)  =  sin  a;'  cos  y  -f-  cos  a;'  sin  y , 
cos(a;f + y)  =  cosa;'  cosy  —  sin  x'  sin  y. 

Hence,  it  follows  that  Formulas  [4]  and  [5]  are  universally 
true.  For  they  have  been  proved  true  for  any  two  acute 
angles,  and  also  true  when  one  of  these  angles  is  increased  by 
90° ;  hence  they  are  true  for  each  repeated  increase  of  one  or 
the  other  angle  by  90°,  and  therefore  true  for  the  sum  of  any 
two  angles  whatever. 

By  §27, 

t     ,     N      sin  (x  +  y)      sin  x  cos  y  -f-  cos  x  sin  y 

tan  (x  +  y)  = ) — p^<  =  — y        r-^« 

cos(a?  +  y)      cosa;  cosy  —  sin  a;  siny 

If  we  divide  each  term  of  the  numerator  and  denominator  of 
the  last  fraction  by  cosa;  cosy,  and  again  apply  §  27,  we  obtain 

,     ,     N       tanx  +  tany  rpl 

tan  (x  4-  y)  = , Z__JL.  [6] 

v     rjJ      1  — tanxtany 

In  like  manner,  by  dividing  each  term  of  the  numerator  and 
denominator  of  the  value  of  cot  (a;  -f-  y)  by  sina;  siny,  we  obtain 

.,       ,       N         COtXCOtV  — 1  r^T 

cot(x--Y)  = ' •  L'J 

v     '  JJ       cotx  +  coty 


GONIOMETRY. 


45 


§  32.   Functions  of  the  Difference  of  Two  Angles. 

In  a  unit  circle  (Fig.  29)  let  the  angle  AOB  —  x,  the  angle 
COB  =  y ;  then  the  angle  AOC= 

*-y. 

In  order  to  express  sin  (x  —  y) 
and  cos  (x  —  y)  in  terms  of  the 
sines  and  cosines  of  x  and  y,  draw 
CF±OA,  CD±OB,  DE±  OAy 
DG±  FC prolonged ;  then  CD= 
siny,  OD  =  cosy,  and  the  angle 
DCG=the  angle  EDC=x.  And, 
sin  (x  -  y)  ==  CF=  DE-  CG. 

— — •  =  sin#:     hence,  DE 
OD 

CG 

— —  =  cos  a; ;     hence,   CG  =  cos  a*  X  CD  —  cos  x  siny. 

Therefore,    sin  (x  —  y)  =  sin x  cos  y  —  cos  x  sin y.  [8] 

Again,  cos  (x  -  y)  =  OF==  OE+DG. 

O  F 

--—  =  coso; ;     hence,  OE  =  cos#  X  OZ>  =  cos;r  cosy. 


Fig.  29. 

sin  a;  X  02)  =  sin  x  cosy. 


CD 


sin  x ;      hence,  DG  =  sin  a?  X  OZ)  =  sin  x  sin  y. 


Therefore,    cos  (x  —  y)  =  cos  x  cos  y  +  sin x  sin y. 


[°] 


In  this  proof,  both  x  and  y  are  assumed  to  be  acute  angles ; 
but,  whatever  be  the  values  of  x  and  y,  the  same  method  of 
proof  will  always  lead  to  Formulas  [8]  and  [9],  when  due 
regard  is  paid  to  the  algebraic  signs. 

The  general  application  of  these  formulas  may  be  at  once 
shown  by  deducing  them  from  the  general  formulas  established 
in  §  31,  as  follows  : 

It  is  obvious  that  (x  —  y)-\~y  —  x.  If  we  apply  Formulas 
[4]  and  [5]  to  (x  --  y)  4-  y,  then 


46  TRIGONOMETRY. 


sin  \(x  —  y)  -f  y  \  or  sin  a;  =  sin  (x  —  y)  cosy  -f-  cos  {x  —  y)  siny, 
cos  5  (a;  —  y)-\-y\  or  cos#  —  cos  (a;  —  y)  cosy  —  sin  (x  —  y)  siny. 

Multiply  the  first  equation  by  cosy,  the  second  by  siny, 

sin x  cosy  —      sin  [x  —  y)  cos2y  -f  cos  (x  —  y)  sin y  cosy, 
cos#  siny  =  —  sin  (x  —  y)  sin2y  -f  cos (x  —  y)  siny  cosy ; 

whence,  by  subtraction, 

sin  z  cosy  —  cos  x  sin  y  =  sin  (x  —  y)  (sin2y  -f-  cos2y). 

But  sin2y  -f  cos2y  =  1 ;  therefore,  by  transposing, 

sin  (#  —  y)  =  sinx  cosy  -  cos x  siny. 

Again,  if  we  multiply  the  first  equation  by  sin  y,  the  second 
equation  by  cosy,  and  add  the  results,  we  obtain,  by  reducing, 

cos  (x  —  y)  =  cos  x  cos  y  -f  sin  x  sin  y. 

Therefore,  Formulas  [8]  and  [9],  like  [4]  and  [5],  from  which 
they  have  been  derived,  are  universally  true. 

From  [8]  and  [9],  by  proceeding  as  in  §  31,  we  obtain 


tan(x      j)      1  +  tanxtany 


[10] 


.j         s      cotxcoty  +  1        >  ni 

cot  (x  —  y)  =  — J—- —  L11 

v       #y       coty  —  cotx 

Formulas  [4] -[11]  may  be  combined  as  follows: 
sin  (a;±y)  =  sin x cosy  ±  cos x siny, 
cos  (x  ±  y)  =  cos  a;  cosy  =F  sin  x  siny, 

.,  N       tan  x  -fc  tan  ?/ 

tan  (a;  db  y)  =  ; — — *-i 

*'       ln=tan#tany 

.  /    v   cot#  coty  =F  1 

cot  (a?  ±  y)  =  — ^— - — 

*'   coty  db  cot  a; 


GONIOMETRY.  47 


§33.   Functions  of  Twice  an  Angle. 

If,  in  Formulas  [4] -[7],  y  ~  x,  they  become  : 

sin  2x  =  2sinxcosx.     [12]  cos2x  =  cos2x  —  sin2x.    [13] 

0  2tanx  --.,  .„        cot2x  —  1  ric1 

tan2x  =  - — — -=-■       [141  cot2x  =  -= — - — .  [15] 

1  — tan2x        L     J  2cotx 

By  these  formulas  the  functions  of  twice  an  angle  are  found 
when  the  functions  of  the  angle  are  given. 

§  34.   Functions  of  Half  an  Angle. 

Take  the  formulas 

cos2  x  -f  sin2#  =  1  [1] 

cos2  x  —  sin2  x  =  cos  2x  [13] 

Subtract,  2  sin2  x  =  1  —  cos  2  a; 

Add,  2  cos2#  =  1  +  cos  2  x 

Whence 


11  —  cos2#                        ,      fl  +  cos2# 
sin x  =  ±  <*/ »         cos x  =  d=  a)  — — 

These  values,  if  z  is  put  for  2x,  and  hence  \z  for  re,  become 

*t\%-±£=*B&.    [16]  coSi2  =  ±iJ^!i.    [17] 

Hence,  by  division  (§  27), 


tan}i  =  ±Jj^5!i    [18]  cot^  =  ±jLt^^,     [19] 

M  +  cosz  J  \1  — cosz 

By  these  formulas  the  functions  of  half  an  angle  may  be 
computed  when  the  cosine  of  the  entire  angle  is  given. 

The  proper  sign  to  be  placed  before  the  root  in  each  case 
depends  on  the  quadrant  in  which  the  angle  \z  lies.     (§  24.) 

Let  the  student  show  from  Formula  [18]  that 

tan  \  B  =J— -•    (See  page  19,  Note  2.' 
\c-\-a 


48  TRIGONOMETRY. 


§  35.   Sums  and  Differences  of  Functions. 

From  [4],  [5],  [8],  and  [9],  by  addition  and  subtraction : 

sin  (x  +  y)  +  sin  (x  —  y)  =      2  sin  x  cos  y , 

sin  (x  +  y)  —  sin  (x  —  y)=      2  cos  x  sin  3/, 

cos  (x  -f-  y)  +  cos  (x  —  y)~      2  cos  2;  cos  y, 

cos  (#  -f"  y)  —  cos  (#  —  y)  =  —  2  sin  x  sin  y ; 

or,  by  making  x-{-y  =  A,  and     x  —  y  =  B, 

and  therefore,  #  =  J  (A  +  -B),  and    y  =  -J  (J.  —  i?), 

sin  A  +  sin  B  =  2  sin  }  (A  +  B)  cos  £  (A  -B).  [20] 
sin  A  -  sinB  =  2cosi(A  +  B)sin  i(A  -  B).  [21] 
cos  A  +  cosB  =  2cos|(A  +  B)cos  £(A  -  B).  [22] 
cos  A  -  cosB  =  -  2  sin  J(A  +  B)  sin  1  (A  -  B).  [23] 
From  [20]  and  [21],  by  division,  we  obtain 

sin^  +  sini?      tan i {A  +  B)cotuA  _  B)  . 


sin  A  —  sin  B 
or,  since  cot -J  (-4  —  B) 


tam$(4— B) 

sin  A  -f  sin  B  _  tan^-  (A  +  B) 
sin  A  —  sin  B     tan  i  (A  —  B)' 

Exercise  XL 


[24] 


1.  Find  the  value  of  sin (x-\-y)  and  cos^-f-y),  when  sinrr 
=  |,  cosa;  =  J,  siny  =  ^,  cosy  =  ff . 

2.  Find  sin  (90° -y)  and  cos  (90° -y)  by  making  ar=90° 
in  Formulas  [8]  and  [9]. 

Find,  by  Formulas  [4] -[11],  the  first  four  functions  of: 

3.  90° +  y.  8.   360° -y.  13.   -y. 

4.  180° -y.  9.   360° +  y.  14.   45° -y. 

5.  180° +  y.  10.   07-90°.  15.   45° +  y. 

6.  270° -y.  11.   a: -180°.         -  16.    30° +  y. 

7.  270° +  y.  12.   a; -270°.  17.   G0°-y. 


GONIOMETRY.  49 


18.  Find  sin  Sx  in  terms  of  sin  x. 

19.  Find  cos  Sx  in  terms  of  cos  a;. 

20.  Given  tan|-a;  =  1 ;  find  cos  x. 

21.  Given  cot  \  x  =  V3  ;  find  sin  a;. 

22.  Given  sins;  =  0.2  ;  find  sin^-a;  and  cos|-a;. 

23.  Given  cos  a;  =  0.5  ;  find  cos 2x  and  tan 2 a;. 

24.  Given  tan  45°  =  1 ;  find  the  functions  of  22°  30'. 

25.  Given  sin  30°  =  0.5 ;  find  the  functions  of  15°. 

26.  Prove  that  tan  18°  =  sin  j*j*°  +  sin  )*°. 

cos  33°  +  cos  3° 

Prove  the  following  formulas  : 

27.  sin2a?=,2tan^  .  29.   tan^ar 


l  +  tan2a;  1-f-cosa; 

28.   cos2a;=}~taA  30.    coUa?: 


l-f-tan2a;  1  — cos  as 

31.    sin-Ja;  =t  cos-|-a;  =  Vl  ±  sina;. 

QO    tan  a;  ±  tan  ?/ 

6Z.   — . _£  =  ±  tan  x  tan  y. 

cot  x  ±  cot  y  ** 

33.  tan(45°-o;)  =  1~tan^ 

v      ,       }      1  +  tana; 

If  A,  B,  Care  the  angles  of  a  triangle,  prove  that: 

34.  sin  A-\-  sin  i?  +  sin  C==4cos|-^.  cos-^-i?  cos-J-C. 

35.  cos  A  +  cos  B  +  cos  (7=  1  +  4  sin -J  ^4  sin  %  B  sin  \  C. 

36.  tan  A  +  tani?  -f  tan  C=  tan  A  X  tan  B  X  tan  C. 

37.  Qot%A  +  ooi%B-{-cQt%0=Got%A  X  cot|i?X  cotJC 

Change  to  forms  more  convenient  for  logarithmic  computa- 
tion: 


38. 

cota;  +  tana;. 

43. 

1  +  tana;  tan  y. 

39. 

cot  x  —  tana;. 

44. 

1—  tan  x  tan  y. 

40. 

cot  x  +  tan  y. 

45. 

cota;coty+l. 

41. 

cot  a;--  tan?/. 

46. 

cota^oty  —  1. 

49 

1  —  cos  2  a: 

47. 

tan  a;  +  tan?/ 

1  -{-cos  2  a; 

cot  a;  +  cot  y 

CHAPTER  IV. 

THE  OBLIQUE  TRIANGLE. 

§  36.   Law  of  Sines. 

Let  A,  B,  C  denote  the  angles  of  a  triangle  A B 0  (Figs.  30 
and  31),  and  a,  b,  c,  respectively,  the  lengths  of  the  opposite 
sides. 

Draw  CD±AB,  and  meeting  AB  (Fig.  30)  or  AB  pro- 
duced (Fig.  31)  at  D.     Let  CD  =  h. 


B      A 


In  both  figures,       --  =  sin^4. 


In  Fig. 

30, 

a 

=  sin 

B. 

In  Fig. 

31, 

h 

a 

=  sin 

(180°- 

■B) 

=  sin  B. 

Therefore, 
we  obtain,  by 

whether 
division, 

h  lief 

!  within 

or 

without  the 

triangle, 

a 
b" 

sin  A 

sinB 

[25] 

THE    OBLIQUE    TRIANGLE. 


51 


By  drawing  perpendiculars  from  the  vertices  A  and  B  to 
the  opposite  sides  we  may  obtain,  in  the  same  way, 


B 


C 


sin  A 
sin  0 


Hence  the  Law  of  Sines,  which  may  be  thus  stated  : 
The  sides  of  a  triangle  are  proportional  to  the  sines  of  the 
opposite  angles. 

If  we  regard  these  three  equations  as  proportions,  and  take 
them  by  alternation,  it  will  be  evident  that  they  may  be  writ- 
ten in  the  symmetrical  form, 


a 

sin  ^4. 


sini? 


0 


Each  of  these  equal  ratios  has  a  simple  geometrical  mean- 
ing which  will  appear  if  the  Law  of  Sines  is  proved  as  follows  : 

Circumscribe  a  circle  about  the  triangle  ABO  (Fig.  32), 
and  draw  the  radii  OA,  OB,  00; 
these  radii  divide  the  triangle  into 
three  isosceles  triangles.  Let  R 
denote  the  radius.  Draw  OM 
Jl  BO  By  Geometry,  the  angle 
BOO  =2A;  hence,  the  angle 
BOM=A,  then  BM=Rsm  BOM 
=  RsinA. 

.'.  BO  or  a  =  2Rsm  A. 

In  like  manner,  b  =  2  R  sin  B, 

and    e  =  2  R  sin  O     Whence  we 

obtain 

b 


2R  = 


sin  A      sin  B     sin  O 


That  is :  The  ratio  of  any  side  of  a  triangle  to  the  sine  of  the 
opposite  angle  is  numerically  equal  to  the  diameter  of  the  cir- 
cumscribed  circle. 


52  TRIGONOMETRY. 


§  37.   Law  of  Cosines. 

This  law  gives  the  value  of  one  side  of  a  triangle  in  terms 
of  the  other  two  sides  and  the  angle  included  between  them. 

In  Figs.  30  and  31,       a2  =  h2  +  BB2. 

In  Fig.  30,  BB  =  c-AB; 

in  Fig.  31,  BB  =AB-c; 

in  both  cases,  BBP  =  ABP -2cxAB+<?. 

Therefore,  in  all  cases,  a2  =  h2  -f  ABP-  +  c2-2cxAB. 

Now,  h2  +  AB2  =  b2, 

and  AB  =  b  cos  A. 

Therefore,  a2  =  b2  +  c2  -  2  be  cos  A.  [26] 

In  like  manner,  it  may  be  proved  that 

h2  =  a2  -f-  c2  —  2ac  cos  B, 
c2  =  a2  +  b2-2abcosC. 

The  three  formulas  have  precisely  the  same  form,  and  the 
law  may  be  stated  as  follows : 

The  square  of  any  side  of  a  triangle  is  equal  to  the  sum  of 
the  squares  of  the  other  two  sides,  diminished  by  twice  the 
product  of  the  sides  and  the  cosine  of  the  included  angle, 

§  38.   Law  of  Tangents. 

By  §  36,  a  ;  b  =  sin  A  :  sin  B ; 

whence,  by  the  Theory  of  Proportion, 

a—h      sin  A  —  sin  B 


a-\-b      sin  A  -J-  sin  B 
But  by  [24],  page  48, 

sin  A  —  sin  B  _  tan  £  (A  —  B) 

sin  A  +  sin  B     tan  \  (A  +  B) 
Therefore, 

a  — b     tan  1  (A  —  B)<  1-07 

a  +  b~tan£(A  +  B) 


THE   OBLIQUE    TRIANGLE.  53 

By  merely  changing  the  letters, 

a  —  c^tB.nKA  —  C)         b  —  c_ta,nh(£—C) 
a  +  c      tan*(^+tf)'        b  +  c      tan  *  (jB  +  C)  " 

Hence  the  Law  of  Tangents : 

The  difference  of  two  sides  of  a  triangle  is  to  their  sum  as  the 
tangent  of  half  the  difference  of  the  opposite  angles  is  to  the  tan- 
gent of  half  their  sum. 

Note.    If  in  [27]  b>a,  then  B>A.     The  formula  is  still  true,  but  to 
avoid  negative  quantities,  the  formula  in  this  case  should  be  written 
b  -  a  _  tan  j  (B  -  A) 
b  +  a  ~  tan  £  (B  +  A) 

Exercise  XII. 

1.  What  do  the  formulas  of  §  36  become  when  one  of  the 
angles  is  a  right  angle  ? 

2.  Prove  by  means  of  the  Law  of  Sines  that  the  bisector  of 
an  angle  of  a  triangle  divides  the  opposite  side  into  parts  pro- 
portional to  the  adjacent  sides. 

3.  What  does  Formula  [26]  become  when  A  =  90°?  when 
^1  =  0°?  when  A  =  180°  ?  What  does  the  triangle  become  in 
each  of  these  cases  ? 

Note.  The  case  where  A  =  90°  explains  why  the  theorem  of  \  37  is 
sometimes  termed  the  Generalized  Theorem  of  Pythagoras. 

4.  Prove  (Figs.  30  and  31)  that  whether  the  angle  B  is 
acute  or  obtuse  c  =  a  cos  B  -f  b  cos  A.  What  are  the  two  sym- 
metrical formulas  obtained  by  changing  the  letters?  What 
does  the  formula  become  when  B  =  90°  ? 

5.  From  the  three  following  equations  (found  in  the  last 
exercise)  prove  the  theorem  of  §  37  : 

c  =  a  cos  B  -}-  b  cos  A, 
b  —  a  cos  C  -f-  c  cos  A, 
a  =  b  cos  C  +  c  cos  B. 
Hint.   Multiply  the  first  equation  by  c,  the  second  by  b,  the  third 
by  a ;  then  from  the  first  subtract  the  sum  of  the  second  and  third. 


54 


TRIGONOMETRY. 


6.  Iii  Formula  [27]  what  is  the  maximum  value  of  2  (A—B)? 

7.  Find  the  form  to  which   Formula   [27]   reduces,  and 
describe  the  nature  of  the  triangle,  when 

(i.)  C=  90° ;       (ii.)  A-B  =  90°,  and  B  =  C. 


§  39.   The  Given  Parts  of  an  Oblique  Triangle. 

The  formulas  established  in  §§  36-38,  together  with  the 
equation  A-{-B-\-  C=  180°,  are  sufficient  for  solving  every 
case  of  an  oblique  triangle.  The  three  parts  that  determine 
an  oblique  triangle  may  be : 

I.   One  side  and  two  angles  ; 
II.    Two  sides  and  the  angle  opposite  to  one  of  these  sides ; 

III.  Two  sides  and  the  included  angle ; 

IV.  The  three  sides. 

In  all  cases  let  A,  B,  O  denote  the  angles,  a,  b,  c  the  sides 
opposite  these  angles  respectively. 


§  40.   Case  I. 

Given  one  side  a,  and  two  angles  A  and  B;  find  the  remain- 
ing parts  C,  b,  and  c. 
1.    C=  180°  -(A  +  JB). 
o     b 


sin 


3.    c-  = 


B 

a     sin  A 
c      sin_(7 

mi 


\b 


a&mB 

sin  A 

a  sin  O 


sin  A 


X  sin  B. 


X  sin  C. 


sin  A       sin  A 
Example,   a  =  24.31,  A  =  45°  18',  B  =  22°  11'. 
The  work  may  be  arranged  as  follows : 
loga  =  1.38578 
cologsin  A  =  0.14825 
logsin.ff  =  9.57700 


a=    24.31 
A^   45°  18' 
B=    22°  11' 
A  +  B=   67°29' 

<7=  112°  31' 


log&  =  1.11103 
b  =  12.913 


1.38578 
0.14825 
9.96556 
log  c  =  1.49959 
c  =  31.593 


log  sin  C^- 


THE    OBLIQUE    TRIANGLE.  55 


Exercise  XIII. 

**1.   Given  a  =  500,         .4  =  10°  12',    -5  =  46°  36'; 

find  C=  123°  12',   b  =  2051.48,    c  =  2362.61. 
A    2.   Given  a  =  795,         J.  =  79°  59',    j?  =  44°41'; 

find  C=  55°  20',     b  =  567.688,    c  =  663.986. 
*3.   Given  a  =  804,         A  =  99°  55',    ^  =  45°r; 

find  ff=  35°  4',       5  =  577.313,    c  =  468.933. 
^4.   Given  a  =  820,         ^4  =  12°  49',    J5  =  141°  59'; 

find  C=  25°  12',     5  =  2276.63,    c  =  1573.89. 
>-5.   Given  <?  =  1005,       4  =  78°  19',    .£  =  54°  27'; 

find  (7=  47°  14',     a  =  1340.6,      &  =  1113.8. 
^6.   Given  6  =  13.57,      £  =  13°  57',     (7=  57°  13'; 

find  .4  =  108°  50',  a  =  53.276,      c  =  47.324. 
A  7.   Given  a  =  6412,       .4  =  70°  55',     C=52°9'; 

find^  =  56°56',     6  =  5685.9,      c  =  5357.5. 
>^  8.   Given  b  =  999,         ^  =  37°  58',     (7=  65°  2' ; 

find^  =  77°,  a  =  630.77,      c  =  929.48. 

J-  9.  In  order  to  determine  the  distance  of  a  hostile  fort  A 
from  a  place  i?,  a  line  BC  and  the  angles  ABC  and  J5C.4 
were  measured,  and  found  to  be  322.55  yards,  60°  34',  and 
56°  10',  respectively.     Find  the  distance  AB. 

10.  In  making  a  survey  by  triangulation,  the  angles  B  and 
C  of  a  triangle  ABC  were  found  to  be  50°  30'  and  122°  9', 
respectively,  and  the  length  BC  is  known  to  be  9  miles.  Find 
AB  and  AC. 

11.  Two  observers  5  miles  apart  on  a  plain,  and  facing 
each  other,  find  that  the  angles  of  elevation  of  a  balloon  in 
the  same  vertical  plane  with  themselves  are  55°  and  58°, 
respectively.  Find  the  distance  from  the  balloon  to  each 
observer,  and  also  the  height  of  the  balloon  above  the  plain. 

12.  In  a  parallelogram  given  a  diagonal  d  and  the  angles 
x  and  y  which  this  diagonal  makes  with  the  sides.  Find  the 
sides.  Compute  the  results  when  d=  11.237,  #=19°  1',  and 
y  =  42°54. 


56  TRIGONOMETRY. 


13.  A  lighthouse  was  observed  from  a  ship  to  bear  N.  34°  E. ; 
after  sailing  due  south  3  miles,  it  bore  N.  23°  E.  Find  the  dis- 
tance from  the  lighthouse  to  the  ship  in  both  positions. 

Note.  The  phrase  to  bear  N.  34°  E.  means  that  the  line  of  sight  to 
the  lighthouse  is  in  the  north-east  quarter  of  the  horizon,  and  makes, 
with  a  line  due  north,  an  angle  of  34°. 

14.  In  a  trapezoid  given  the  parallel  sides  a  and  b,  and  the 
angles  x  and  y  at  the  ends  of  one  of  the  parallel  sides.  Find 
the  non-parallel  sides.  Compute  the  results  when  a  =  15, 
&«7,  #  =  70°,  y  =  40°. 

Solve  the  following  examples  without  using  logarithms : 

15.  Given  b  =  7.07107,    A  =  30°,    C=  105°  ;  find  a  and  c. 

16.  Given  <?  =  9.562,        ^4  =  45°,  ^  =  60°;    find  a  and  b. 

17.  The  base  of  a  triangle  is  600  feet,  and  the  angles  at  the 
base  are  30°  and  120°.     Find  the  other  sides  and  the  altitude. 

18.  Two  angles  of  a  triangle  are,  the  one  20°,  the  other  40°. 
Find  the  ratio  of  the  opposite  sides. 

19.  The  angles  of  a  triangle  are  as  5  :  10  :  21,  and  the  side 
opposite  the  smallest  angle  is  equal  to  3.  Find  the  other 
sides. 

20.  Given  one  side  of  a  triangle  equal  to  27,  the  adjacent 
angles  equal  each  to  30°.  Find  the  radius  of  the  circum- 
scribed circle.     (See  §  36,  Remark.) 


§41.   Case  II. 

Given  two  sides  a  and  b,  and  the  angle  A  opposite  to  the 
side  a ;  find  the  remaining  parts  B,  C,  c. 

This  case,  like  the  preceding  case,  is  solved  by  means  of 

the  Law  of  Sines. 

0.  sini?      b     ,-,       f  .     -n      b  sin  A  . 

Since  - — -  =  — ,    tnereiore    sin  B  = > 

sin  A      a  a 

C=  180°  -  (A  +  B). 

A     j  c      sin  C     ,i        n  a  sin  O 

And  since       -  =  - 7,    therefore    c  =  — : — —  • 

a      sin  A  sin  A 


THE   OBLIQUE    TRIANGLE.  57 

When  an  angle  is  determined  by  its  sine  it  admits  of  two 
values,  which  are  supplements  of  each  other  (§  28) ;  hence, 
either  value  of  B  may  be  taken  unless  excluded  by  the  con- 
ditions of  the  problem. 

If  a>  b,  then  by  Geometry  A  >  B,  and  B  must  be  acute 
whatever  be  the  value  of  A ;  for  a  triangle  can  have  only 
one  obtuse  angle.  Hence,  there  is  one,  and  only  one,  triangle 
that  will  satisfy  the  given  conditions. 

If  a  =  b,  then  by  Geometry  A  =  B ;  both  A  and  B  must  be 
acute,  and  the  required  triangle  is  isosceles. 

If  a  <  b,  then  by  Geometry  A  <  B,  and  A  must  be  acute 

in  order  that  the  triangle 

may  be  possible.     If  A  is  ^S 

acute,   it  is  evident   from 

Fig.  33,  where  Z.  BA  C=A,  /^ 

AC  =  b,    CB  =  CB'=  a,  J^  / 

that  the  two  triangles  A  CB  ^^  y 

and  ACB'  will  satisfy  the      //         ■.,     / 

given  conditions,  provided  ^  TP^Z         P 

a  is  greater  than  the  per-  ^ 

pendicular    CP;    that    is, 

provided  a  is  greater  than  b  sin  A  (§  10).     The  angles  ABC 

and  AB'C  are  supplementary   (since  Z.  ABC—  Z.  BB'C)  ; 

they  are  in  fact  the  supplementary  angles  obtained  from  the 

formula  •     r>      b  sin  A 

sin  Jd  = 

a 

If,  however,  a  —  b  sin  A  =  CP  (Fig.  33),  then  sin  B  =  1 , 
B  =  90°,  and  the  triangle  required  is  a  right  triangle. 

If  a<b sin  A,  that  is,  <  CP,  then  sini?>l,  and  the  tri- 
angle is  impossible. 

These  results,  for  convenience,  may  be  thus  stated : 

Ifa>&,     or      a  =  b,  or  if    a  =  b  sin  A,  One  solution. 

If  a  <  b,     but  >  b  sin  A,    and    A  <  90°,        Two  solutions. 

If«<6      and^L>90°,      or  if    a  <  b  sin  A  and  A  <  90°, 

No  solution. 


58 


TRIGONOMETRY. 


The  number  of  solutions  can  often  be  determined  by  inspec- 
tion. If  there  is  any  doubt,  it  may  be  removed  by  computing 
the  value  of  b  sin  A. 

Or  we  may  proceed  to  compute  log  sin  B.  If  log  sin  B  =  0, 
the  triangle  required  is  a  right  triangle.  If  log  sin  i?>0,  the 
triangle  is  impossible.  If  log  sin  B<0,  there  is  one  solution 
when  a>5  ;  there  are  two  solutions  when  a<b. 

When  there  are  two  solutions,  let  B\  C\  c\  denote  the  un- 
known parts  of  the  second  triangle  ;  then, 

&  =  180° -B,  C -  180° -(A  +  B')=B-A, 
a  sin  C1 


c  = 


sin  A 


Examples, 
20,  ^4  =  106c 


find   the  remaining 


1.  Given    a  =16,  b 
parts. 

In  this  case  a  <  b,  and  A  >  90°  ;  therefore  the  triangle  is  impossible. 

2.  Given   a  =  36,  b  =  80,  A  =  30° ;    find   the   remaining 
parts. 

Here  we  have  b  sin  A  —  80  X  $  —  40;   so  that  a  <  b  sin  A,  and  the 

triangle  is  impossible. 

3.  Given  a  =  72630,  b  =  117480,  ^4  =  80°0'  50";  find^,C,c. 


a  -  72630 
6-117480 
A  =  80°  0'  50" 


colog  a  =  5.13888 

log  b  =  5.06996 

log  sin  A  -  9.99337 


Here  log  sin  B>0. 
:.  no  solution. 


log  sin  B  =  0.20221 
4.   Given  a  =  13.2,  6  =  15.7,  ^4  =  57°  13' 15.3";  find  B,C,c> 


a  - 13.2 
b  -  15.7 
4  =  57°  13' 15.3' 


Here  log  sin  B  =  0, 
■\  a  right  triangle. 


colog  a  -  8.87943 

log  b  =  1.19590 

log  sin  A  =  9.92467 

log  sin  B  =  0.00000 
5=90° 
.-.  C  -32°  46'  44.7" 


c  =  b  cos  A 

log  6  =  1.19590 

log  cos  4-  9.73352 

log  c  -  0.92942 
e-8.5 


THE   OBLIQUE   TRIANGLE. 


59 


5.    Given  a 

a  =  767 
&  =  242 
A  =  36°  53'  2" 


767,  5-242,  ^  =  36°53'2 
colog  a  =7.11520 
log  b  =  2.38382 
log  sin  A  =  9.77830 

log  sin  B  =  9.27732 

B  =10°  54'  58" 
.-.  (7=132°  12' 0" 


;  find  JB,  C,  c. 
log  a  =  2.88480 
log  sin  O  =  9.86970 
cologsinJ.  =  0.22170 

log  c  =  2.97620 


Here  a  >  5, 

and  log  sin  B  <  0.  5=10°  54'  58  " 

.°.  one  solution. 

6.    Given  a  =  177.01,  6  =  216.45,  ^  =  35°36'20";  find  the 
other  parts. 

a  =177.01 
b  =  216.45 
A  =  35°  36'  20" 


Here  a<b, 

and  log  sin  B  <  0, 

,*.£u;o  solutions. 


colog  a  =  7.75200 
log  b  =  2.33536 

log  sin  ,1=9.76507 

log  sin  B  =  9.85243 

B  =  45°  23'  28" 
or  134°  36'  32" 
.-.  C  =  99°  0'  12" 
or  9°  47'  8" 


log  a; 

cologsinJ.; 

logsin(7  = 


:  2.24800 
:  0.23493 
:  9.99462 


log  c  =  2.47755 


2.24800 
0.23493 
9.23034 


1.71327 


c=  300.29  or  51.674 


1.    Determine 
lowing  cases : 

(ii.)a  = 

(iii.)  a  = 

(iv.)  a  = 
(v.)a  = 

(vi.)a  = 
<    2.    Given  a  = 

find^  = 
Given  a  = 

findi?  = 
Given  a  = 

find  .5  = 
Given  a  = 

find  .4  - 


Exercise  XIV. 
the  number  of  solutions  in  each  of  the  fol- 


80, 

50, 

40, 

13.4, 

70, 

134.16, 

840, 


6  = 
6  = 
6  = 
6  = 
6  = 
&  = 


100, 

100, 

100, 

11.46, 

75, 

84.54, 


A  -  30°. 
A  =  30°. 
^  =  30°. 
.4  =  77°  20'. 
A  =  60°. 
.5  =  52°  9' 11". 


485, 


.4  =  21°  31' 


3. 


4. 


9.399, 
57°  23' 40", 
91.06, 
41°  13', 
55.55, 
54°  31' 13", 


b 

(7=  146°  15' 26",   c=  1272.18. 
6  =  9.197,  ,4  =  120°  35'; 

<7=2°1'20',  c  =  0.38525. 

6  =  77.04,  ^  =  51°  9'  6" 

(7=  87°  37' 54",     e  =  116.82. 


I  & 


6  =  66.66,        ^ 


44' 40 


(7=  47°  44' 7",  c  =  50.481. 


60  TRIGONOMETRY. 


6.  Given  a  =  309,  6  =  360,  .4  =21°  14' 25"; 

find  B  =  24°  57' 54",    C=  133°47'41",  c  =  615.67, 
£*=  155°  2'  6",      C'=  3°  43'  29",      c'=  55.41. 

7.  Given  a  =  8.716,  6  =  9.787,  J.  =  38°  14'  12"; 

find  £  =  44°  1'  28",      (7=  97°  44' 20",    (7  =  13.954, 
5'=  135°  58' 32",  C"=  5°  47'  16",      c'=  1.4203. 

8.  Given  a  =4.4,     6  =  5.21,  A  =  57°  37'  17"; 

find  ^  =  90°,     (7=  32°  22' 43",   c  =  2.79. 

9.  Given  a  =  34,  6  =  22,  B  =  30°  20' ; 

find  A  =  51°  18'  27",       C=  98°  21'  33",     c  =  43.098, 

^4'=  128°  41'  33",    C'=  20°  58'  27",     <?'=  15.593. 

10.    Given  6  =  19,  c  =  18,  C=  15°  49' ; 

find  £  =  16°  43' 13",     ^  =  147°  27' 47",  a  =  35.519, 

.£'=163°  16'  47",  .4'=  0°  54' 13",       a'=  1.0415, 

11.  Given  a  =75,  6  =  29,  .#=16°  15' 36";  find  the  differ- 
ence between  the  areas  of  the  two  corresponding  triangles. 

12.  Given  in  a  parallelogram  the  side  a,  a  diagonal  d,  and 
the  angle  A  made  by  the  two  diagonals ;  find  the  other  diag- 
onal. 

Special  case:  a  =  35,  d=63,  A  =  21°  36' 30". 

§  42.   Case  III. 

Given  two  sides  a  and  6  and  the  included  angle  C ;  find  the 
remaining  parts  A,  B,  and  c. 

Solution  I.  The  angles  A  and  B  may  both  be  found  by 
means  of  Formula  [27],  §  38,  which  may  be  written 

tan*(^  -B)  =^Zl|  x  tan*(^  +  £). 
a-\-b 

Since  }  {A  +  B)  =  $  (180°  -  O),  the  value  of  i(A  +  B)  is 
known ;  so  that  this  equation  enables  us  to  find  the  value  of 
l(A~  B).     We  then  have 

i{A  +  B)  +  \{A-B)=*At 

and  l(A  +  B)-l  {A  -  B)  =  B. 


THE    OBLIQUE    TKIANGLE. 


61 


After  A  and  B  are  known,  the  side  c  may  be  found  by  the 
Law  of  Sines,  which  gives  its  value  in  two  ways,  as  follows : 


a  sin  (7 


or    c  = 


b  sin  C 


sin  A'  sin  i? 

Solution  II.    The  third  side  c  may  be  found  directly  from 
the  equation  (§  37) 

c  =  -)/(*  + P-SffotmC;. 

and  then,  by  the  Law  of  Sines,  the  following  equations  for 
computing  the  values  of  the  angles  A  and  B  are  obtained  ; 


sin  A  —  a  X 


sin  C 


sin  B  =  b  X 


sin  C 


Solution  III.    If,  in  the  triangle  ABC  (Fig.  34),  BD  is 
drawn    perpendicular    to  the   side 
AC,  then 

BD  __      BD 
AD     AC- DC 
Now  BD  =  asmC     (§10), 

and  DC  =  a  cos  C 


•tan  A 


tan  A 


a  sin  C 


b  —  a  cos  C 
By  merely  changing  the  letters, 

b  sin 


tani?  = 


0 


a  —  b  cos  C 

It  is  not  necessary,  however,  to  use  both  formulas.  When 
one  angle,  as  A,  has  been  found,  the  other,  B,  may  be  found 
from  the  relation  A  +  B-{-  C=  180°. 

When  the  angles  are  known,  the  third  side  is  found  by  the 
Law  of  Sines,  as  in  Solution  I. 

Note.  When  all  three  unknown  parts  are  required,  Solution  I.  is  the 
most  convenient  in  practice.  When  only  the  third  side  c  is  desired,  Solu- 
tion II.  may  be  used  to  advantage,  provided  the  values  of  a2  and  b2  can 
be  readily  obtained  without  the  aid  of  logarithms.  But  Solutions  II. 
and  III.  are  not  adapted  to  logarithmic  work. 


62 


TRIGONOMETRY. 


1.   Given   a  =748,   b 
and  c. 


Examples. 

=  375,    C=  63°  35' 30";   find  A,  B, 


a  +  b  = 

1123 

a  —  b  = 

373 

(A+B)  = 

116° 

21'  30" 

t(A+B)  = 

58° 

12'  15" 

\(A-B)  = 

28° 

10'  52" 

A  = 

86° 

23'    7" 

B=> 

30° 

1'23''  1 

log  (a -6)  =  2.57171 

colog(a  +  b)  =  6.94961 

log  tan  \  (A+B)  =  0.20766 

log  tan  |(4-^)- 9.72898 

I  (A-B)  =  28°  10' 52" 


log  b  =  2.57103 

log  sin  (7=9.95214 

colog  sin  j5  =  0.30073 

lose  =  2.82690 


log 


.671.27 


Note.  In  the  above  Example  we  use  the  angle  B  in  finding  the  side 
c,  rather  than  the  angle  A,  because  A  is  near  90°,  and  therefore  its  sine 
should  be  avoided. 

2.    Given  a  =  4,  c  =  6,  B  =  60°;  find  the  third  side  b. 
Here  Solution  II.  may  be  used  to  advantage.     We  have 
b  =  Va?  +  d*-  2  ac  cos  B=  Vl6  +  36  -  24  =  V28 ; 
log  28  -  1.44716,     log  V28  =  0.72358,     V28  -  5.2915 ; 
that  is,       6  -  5.2915. 


I    3- 


Exercise  XV. 

Given  a  =  77.99, 

b  =  83.39,      C- 

=  72°  15';  [§38,N. 

find  A  =  51°  15', 

B  =  56°  30', 

c  =  95.24. 

Given  b  =  872.5, 

c  =  632.7, 

^  =  80°; 

find  .£  =  60°  45', 

C=--  39°  15', 

a  =984.83. 

Given  a=  17, 

6  =  12, 

(7=  59°  17'; 

find  A  =  77°  12'  53", 

£  =  43°  30'  7", 

c  =  14.987. 

Given  b  =  V5, 

c=  V3, 

^  =  35°  53'; 

find  .£  =  93°  28'  36", 

C=  50°  38'  24", 

c*  =  1.313. 

Given  a  =  0.917, 

6  =  0.312, 

C=  33°  7' 9"; 

find  A  =  132°  18' 27", 

£=14°  34'  24", 

c  =  0.67748. 

Given  a  =  13.715, 

c=  11.214, 

j5=15°22'36" 

find^  =  118°55'49", 

C=45°41'35'\ 

6  =  4.1554. 

Given  b  =  3000.9, 

c  =  1587.2, 

^  =  86°  4'  4"; 

find  .£  =  65°  13' 51", 

C=  28°  42'  5", 

a  =  3297.2. 

THE    OBLIQUE    TRIANGLE.  G3 


8.  Given  a  =  4527,  6  =  3465,  tf=66°6'27"; 

find  A  =  68°  29'  15",    B  =  45°  24r  18",  c  =  4449. 

9.  Given  a  =  55.14,  6  =  33.09,  C=  30°  24'; 

find  A  =  117°  24'  33",  j?  =  32°  11'  27",  c  =  31.431. 

10.  Given  a  =  47.99,         6  =  33.14,         C=  175°  19'  10" ; 

find  A  =  2°  46'  8",   ^  =  1°54'42",   c  =  81.066. 

11.  If  two  sides  of  a  triangle  are  each  equal  to  6,  and  the 
included  angle  is  60°,  find  the  third  side. 

12.  If  two  sides  of  a  triangle  are  each  equal  to  6,  and  the 
included  angle  is  120°,  find  the  third  side. 

13.  Apply  Solution  I.  to  the  case  in  which  a  =  b  or  the 
triangle  is  isosceles. 

14.  If  two  sides  of  a  triangle  are  10  and  11,  and  the  in- 
cluded angle  is  50°,  find  the  third  side. 

15.  If  two  sides  of  a  triangle  are  43.301  and  25,  and  the 
included  angle  is  30°,  find  the  third  side. 

16.  In  order  to  find  the  distance  between  two  objects  A 
and  B  separated  by  a  swamp,  a  station  C  was  chosen,  and  the 
distances  CA  =  3825  yards,  CB  =  3475.6  yards,  together  with 
the  angle  ACB  =  62°  31',  were  measured.  Find  the  distance 
from  A  to  B. 

17.  Two  inaccessible  objects  A  and  B  are  each  viewed 
from  two  stations  C and  D  562  yards  apart.  The  angle  ACB 
is  62°  12',  BCD  41°  8',  ABB  60°  49',  and  ABC  34°  51'; 
required  the  distance  AB. 

18.  Two  trains  start  at  the  same  time  from  the  same  station, 
and  move  along  straight  tracks  that  form  an  angle  of  30°,  one 
train  at  the  rate  of  30  miles  an  hour,  the  other  at  the  rate  of 
40  miles  an  hour.  How  far  apart  are  the  trains  at  the  end 
of  half  an  hour? 

19.  In  a  parallelogram  given  the  two  diagonals  5  and  6, 
and  the  angle  that  they  form  49°  18'.     Find  the  sides. 

20.  In  a  triangle  one  angle  =  139°  54',  and  the  sides  form- 
ing the  angle  have  the  ratio  5:9.    Find  the  other  two  angles. 


64  TRIGONOMETRY. 


§  43.   Case  IV. 

&iven  the  three  sides  a,  b,  c ;  find  the  angles  A,  £,  C. 
The  angles  may  be  found  directly  from  the  formulas  estab« 
lished  in  §  37.    Thus,  from  the  formula 

a2  =  b2  +  c2-2bccosA 

b2  4-  c2  —  a2 

we  have  cos  A  = '.     '    T 

2bc 

From  this  equation  formulas  adapted  to  logarithmic  work 
are  deduced  as  follows : 

For  the  sake  of  brevity,  let  a  +  b  -f-  c  =  2  s ;  then  b  -J-  c  —  a 
=  2(s  —  a),    a  —  b  -f-  c  =  2 {s  —  b),  and  a  +  b  —  c  =  2 (s  —  c). 

Then  the  value  of  1  —  cos  A  is 

1      b2  +  c2  -  a2  _2bc  -b2  -  <?  +  a2  _a2  -  (b-  cf 
2bc  2bc  2bc 

_  (a  +  b  -  c)(a -  b  +  c)  __2(s  -b)(s -  c)  . 
2fo  6c  J 

and  the  value  of  1  -f-  cos  A  is 

1      y  +  c»-q»^26c  +  y  +  c>-a'.=  (&  +  g)'-a' 
26c  2bc  2bc 

_(b  +  c  +  a)(b  +  c  -  a)  ^2s(s  -  a) 
2bc  be 

But  from  Formulas  [16]  and  [17],  §  34,  it  follows  that 

1  —  cos  A  =  2sm2$A,  and  1  +  cos  A  =2cos2j^4. 

;    0  0^2  1   a       2  (s  —  b)  (s  —  c)          n0        o  1    /t        2<s(s  — a) 
.' .  2  sin'  £  ^4  =  — i -^ *,  and  2  cos"5  2  ^1  =  — *- '> 


whence 


and  therefore 


■inU^C- b>fr- °>,  [28] 

V8^  P»] 


cos  1 A 


tenU=^1^ri-  rsoi 


s  (s  —  a) 


THE   OBLIQUE   TRIANGLE.  G5 

By  merely  changing  the  letters, 

•       i    -n  l(S  —  °)  (S  —  C)  •       i   n  l(S  —  a)  (S  ~  ^) 


(s  —  a)  (5  —  /;) 


(s  —  b)  \      s  (s  —  c) 

There  is  then  a  choice  of  three  different  formulas  for  finding 
the  value  of  each  angle.  If  half  the  angle  is  very  near  0°, 
the  formula  for  the  cosine  will  not  give  a  very  accurate  result, 
because  the  cosines  of  angles  near  0°  differ  little  in  value ;  and 
the  same  holds  true  of  the  formula  for  the  sine  when  half 
the  angle  is  very  near  90°.  Hence,  in  the  first  case  the 
formula  for  the  sine,  in  the  second  that  for  the  cosine,  should 
be  used. 

But,  in  general,  the  formulas  for  the  tangent  are  to  be 
preferred. 

It  is  not  necessary  to  compute  by  the  formulas  more  than 
two  angles ;  for  the  third  may  then  be  found  from  the  equation 

A  +  B  +  C=180°. 

There  is  this  advantage,  however,  in  computing  all  three 
angles  by  the  formulas,  that  we  may  then  use  the  sum  of  the 
angles  as  a  test  of  the  accuracy  of  the  results. 

In  case  it  is  desired  to  compute  all  the  angles,  the  formulas 
for  the  tangent  may  be  put  in  a  more  convenient  form. 

The  value  of  tan  2  A  may  be  written 


tis-a)(s-b)(s-c)    Qr      1        j(s-g)(s~b)(s  -g) 

Hence,  if  we  put 

f(s-a)(s-b)(s-c)_r>  [31-| 

we  have  tan 2  A  = •  [32] 

s  —a 


GG 


TRIGONOMETRY. 


In  like  manner, 

tan  i  B 


-V 


tan  k  C 


Examples. 


1.    Given  a  =  3.41,    b  =  2.G0,    c  =  1.58;  find  the  angles. 
Using  Formula  [30],  and  the  corresponding  formula  for  tan  \  B,  wt 
may  arrange  the  work  as  follows : 


a  = 

=  3.41 

6  = 

=  2.60 

c  = 

=  1.58 

2s- 

=  7.59 

s  = 

=  3  795 

s 

—  a  = 

=  0.385 

a 

-b  = 

- 1.195 

s 

—  c  • 

=  2.215 

colog  s  =  9.42079 

colog(s-  a)  =  0.41454 

log  («-  b)  =  0.07737 

log  (s-c)  =  034537 

2)0.25807 

log  tan  J  J.  =  0.12903 

%A  =    53°  23' 20" 
"  A  =  106°  46'  40" 


colog  «=  9.42079-10 

log(«-a)  =  9.58546-10 

colog  (s-  b)  =  9.92263-10 

log  (s  -  c)  =  0.34537 

2)19.27425  -  20 

log  tan  \B  =  9.63713-10 

\B~  23°  26'  37" 

,5=  46°  53'  14" 


.:A  +  B  =  153°  39'  54",  and  (7-  26°  20'  6". 

2.  Solve  Example  1  by  finding  all  three  angles  by  the  use 
of  Formulas  [31]  and  [32]. 

Here  the  work  may  be  compactly  arranged  as  follows,  if  we  find 
log  tan  \  A,  etc.,  by  subtracting  log(s  —  a),  etc.,  from  logr  instead  of 
adding  the  cologarithm  :  • 


a  =  3.41 

log(s-  a)  =  9.58546 

log 

tan  \A  -  10.12903 

b  -  2.60 

log («-  b)  =  0.07737 

log 

tan  \B=    9.63713 

c  =  1.58 

log(«-c)  =  0.34537 
colog  s  =  9.42079 

log 

tan  \  C  =    9.36912 

is  =7.59 

M-    53°  23' 20" 

s  =  3.795 

log  r2  =  9.42899 

\B=    23°  26'  .".7" 

a  =  0385 

logr  =9.71450 

|0-    13°  10'   3" 

b  =  1.195 

4  -  106°  46'  40" 

c  =  2.215 

J?=    46°  53' 14" 

s  -  7.590  (pre 

►of). 

C  =    26°  20'    G" 

Rroof,  A  +  B  +  C  -  180°    V    0" 

Note.  Even  if  no  mistakes  are  made  in  the  work  the  sum  of  the 
three  angles  found  as  above  may  differ  very  slightly  from  180°  in  conse- 
quence of  the  fact  that  logarithmic  computation  is  at  best  only  a  method 
of  close  approximation.  When  a  difference  of  this  kind  exists  it  should 
be  divided  among  the  angles  according  to  the  probable  amount  of  error 
for  each  angle. 


^HE    OBLIQUE    TRIANGLE. 


Exercise  XVI. 

Solve  the  following  triangles,  taking  the  three  sides  as  the 
given  parts : 


1 

a 

6. 

c 

A 

B 

1 
Q 

/- 

51 

66 

20 

38°  52'  48" 

126°  52' 12" 

14°  15' 

A- 

2 

78 

101 

29 

32°  10' 54" 

136°  23' 50" 

11°  25'  16" 

3 

111 

145 

40 

27°  20'  32" 

143°  7' 48" 

9° 31 '40" 

4 

21 

26 

31 

42°  6' 13" 

56°  6' 36" 

81°  47'  11" 

5 

19 

34 

49 

16°  25'  36" 

30°  24' 

133° 10' 24" 

6 

43 

50 

57 

46°  49'  35" 

57°  59' 44" 

75° 10' 41" 

7 

37 

58 

79 

26°  0'29" 

43°  25'  20" 

110°  34' 11" 

8 

73 

82 

91 

49°  34'  58" 

58°  46' 58" 

71°  38'  4" 

9 

14.493 

55.4363 

66.9129 

8°  20' 

33°  40' 

138° 

10 

y/E 

Vo 

^7 

51°  53' 12" 

59°  31'  48" 

68°  35' 

11.  Given  a  =  6,  5  =  8,  e  =  10;  find  the  angles. 

12.  Given  a  =  6,  5  =  6,  c  =  10 ;  find  the  angles. 

13.  Given  a  =  6,  5  =  6,  <?  =  6  ;     find  the  angles. 

14.  Given  a  =  6,  5  =  5,  c  =  12 ;  find  the  angles. 

15.  Given  a  =  2,  5  =  V6,  c  =  V3  —  1 ;  find  the  angles. 

16.  Given  a  =  2,  5  =  V6,  c==V8+l;  find  the  angles. 

17.  The  distances  between  three  cities  A,  JB,  and  Care  as 
follows :  AB  =  165  miles,  AC=  72  miles,  and  BO=  185  miles. 
i>  is  due  east  from  A.  In  what  direction  is  Cfrom  A  ?  What 
two  answers  are  admissible? 

18.  Under  what  visual  angle  is  an  object  7  feet  long  seen 
by  an  observer  whose  eye  is  5  feet  from  one  end  of  the  object 
and  8  feet  from  the  other  end  ? 

19.  When  Formula  [28]  is  used  for  finding  the  value  of  an 
angle,  why  does  the  ambiguity  that  occurs  in  Case  II.  not 
exist  ? 

20.  If  the  sides  of  a  triangle  are  3,  4,  and  6,  find  the  sine 
of  the  largest  angle. 

21.  Of  three  towns  A,  B,  and  Cf  A  is  200  miles  from  B 
and  184  miles  from  C,  B  is  150  miles  due  north  from  C\  how 
far  is  A  north  of  C? 


g8  trigonometry. 

§  44.   Area  of  a  Triangle. 

If  F  denote  the  area  of  the  triangle  ABC  (Fig.  30  or  31, 
page  50),  then,  by  Geometry, 

F=  \c  x  CD. 

By  §  10,  CD  — a  sin  B. 

Therefore,  P  =  \  ac  sinB.  [33] 

And,  in  like  manner, 

F=  $ab  sin  C    and     F=  Ibcs'mA. 

That  is :  TAc  area  o/*  a  triangle  is  equal  to  half  the  product 
of  two  sides  and  the  sine  of  the  included  angle. 

By  Formula  [33]  the  area  of  a  triangle  may  be  found  directly 
when  two  sides  and  the  included  angle  are  given  ;  in  the  other 
cases  the  formula  may  be  used  when  these  parts  have  been 
computed. 

When  the  three  sides  of  a  triangle  are  given,  as  in  Case  IV., 
a  formula  for  its  area  may  be  found  as  follows : 

By  §33,    Bin^B=r2siniJ?Xcoe|JB. 

By  substituting  for  sin  §  B  and  cos  h  B  their  values  in  terms 
of  the  sides  given  in  §  43, 

2     

sin  B  =  —  Vs (s  —  a)  (s  —  b)  (s  —  c). 

By  substituting  this  value  of  sin  B  in  [33], 

r-Vs(s-a)(s-b)(s-c).  [34] 

If  R  denote  (as  in  §  3G)  the  radius  of  the  circumscribed 
circle,  we  have,  from  §  36, 

sinJ5  =  A. 

By  substituting  this  value  of  sin  B  in  [33], 
■n      abc 


4E  ^ 


THE    OBLIQUE    TRIANGLE.  69 

If  r  denote  the  radius  of  the  inscribed  circle,  and  we  divide 
the  triangle  into  three  triangles  by  lines  from  the  centre  of 
this  circle  to  the  vertices,  the  altitude  of  each  of  the  three  tri- 
angles is  equal  to  r.     Therefore, 

P  --  j  r  (a  +  b  +  c)  =  rs.  [3G] 

By  substituting  in  this  formula  the  value  of  F  given  in  [34], 


4 


(s  —  a)(s  —  b)  (s  —  c)m 


whence  ? 
circle. 


,  in  [31]  §  43,  is  equal  to  the  radius  of  the  inscribed 

Exercise  XVII. 
Find  the  area  : 

1.  Given  a  =  4474.5, 

2.  Given  b  =  21.66, 

3.  Given  a  =  510, 

4.  Given  a  =  408, 

5.  Given  a  =  40, 

6.  Given  a  =  624, 

7.  Given  b  =  149, 

8.  Given  a  =  215.9, 

9.  Given  5  =  8, 

10.  Given  a  =7, 

11.  Given  a  =60,    B 
radius  of  the  inscribed  circle. 

12.  Obtain  a  formula  for  the  area  of  a  parallelogram  in 
terms  of  two  adjacent  sides  and  the  included  angle. 

13.  Obtain  a  formula  for  the  area  of  an  isosceles  trapezoid 
in  terms  of  the  two  parallel  sides  and  an  acute  angle. 

14.  Two  sides  and  included  angle  of  a  triangle  are  2416, 
1712,  and  30°  ;  and  two  sides  and  included  angle  of  another 
triangle  are  1948,  2848,  and  150°  ;  find  the  sum  of  their  areas. 

15.  The  base  of  an  isosceles  triangle  is  20,  and  its  area  is 
100  ■+■  V3  ;  find  its  angles. 


>b 

=  2164.5 

,P- 

-116°  30'  20". 

c 

=  36.94, 

A- 

=  66°  4'  19". 

c 

=  173, 

B 

-162°  30' 28". 

b 

=  41, 

c- 

=  401. 

b 

=  13, 

c- 

=  37. 

b 

=  205, 

c  - 

=  445. 

A 

=  70°  42'  30" 

\  .5  =  39°  18'  28". 

c 

=  307.7, 

A- 

=  25°  9' 31". 

c 

=  5, 

A-- 

=  60°. 

c 

=  3, 

A- 

=  60°. 

= 

40°  35'  12", 

area  =  12 ;    find 

the 

70  TRIGONOMETRY. 


Exercise  XVIII. 

1.  From  a  ship  sailing  down  the  English  Channel  the  Eddy- 
stone  was  observed  to  bear  N.  33°  45'  "W. ;  and  after  the  ship 
had  sailed  18  miles  S.  67°  30' W.  it  bore  N.  11°  15'  E.  Find 
its  distance  from  each  position  of  the  ship. 

2.  Two  objects,  A  and  B,  were  observed  from  a  ship  to 
be  at  the  same  instant  in  a  line  bearing  N.  15°  E.  The  ship 
then  sailed  north-west  5  miles,  when  it  was  found  that  A  bore 
due  east  and  B  bore  north-east.  Find  the  distance  from  A 
to  B. 

3.  A  castle  and  a  monument  stand  on  the  same  horizontal 
plane.  The  angles  of  depression  of  the  top  and  the  bottom  of 
the  monument  viewed  from  the  top  of  the  castle  are  40°  and 
80° ;  the  height  of  the  castle  is  140  feet.  Find  the  height  of 
the  monument. 

4.  If  the  sun's  altitude  is  60°,  what  angle  must  a  stick  make 
with  the  horizon  in  order  that  its  shadow  in  a  horizontal 
plane  may  be  the  longest  possible? 

5.  If  the  sun's  altitude  is  30°,  find  the  length  of  the  longest 
shadow  cast  on  a  horizontal  plane  by  a  stick  10  feet  in  length. 

C.  In  a  circle  with  the  radius  3  find  the  area  of  the  part 
comprised  between  parallel  chords  whose  lengths  are  4  and  5 
(Two  solutions.) 

7.  A  and  B,  two  inaccessible  objects  in  the  same  horizontal 
plane,  are  observed  from  a  balloon  at  C  and  from  a  point  B 
directly  under  the  balloon,  and  in  the  same  horizontal  plane 
with  A  and  B.  If  CD  =  2000  yards,  Z  ACD  =  10°  15'  10", 
Z  BCD  =  6°  7'  20",  Z  ADB  =  49°  34'  50",  find  AB. 

8.  A  and  B  are  two  objects  whose  distance,  on  account  of 
intervening  obstacles,  cannot  be  directly  measured.  At  the 
summit  C  of  a  hill,  whose  height  above  the  common  horizontal 
plane  of  the  objects  is  known  to  be  517.3  yards,  Z  ACB  id 
found  to  be  15°  13'  15".  The  angles  of  elevation  of  C  viewed 
from  A  and  B  -are  21°  9' 18"  and  23°  15'  34"  respectively. 
Find  the  distance  from  A  to  B. 


MISCELLANEOUS    PROBLEMS. 

[Selected  by  permission  from  "  Problems  in  Plane  Trigonometry," 
prepared  by  Prof.  C.  J.  White,  of  Harvard  College,  and  published  by 
Charles  W.  Sever,  Cambridge.] 

1.  The  angular  distance  of  any  object  from  a  horizontal 
plane,  as  observed  at  any  point  of  that  plane,  is  the  angle 
which  a  line  drawn  from  the  object  to  the  point  of  observa- 
tion makes  with  the  plane.  If  the  object  observed  be  situated 
above  the  horizontal  plane  (that  is,  if  it  is  farther  from  the 
earth's  centre  than  the  plane  is),  its  angular  distance  from 
the  plane  is  called  its  angle  of  elevation.  If  the  object  be 
below  the  plane,  its  angular  distance  from  the  plane  is  called 
its  angle  of  depression.  These  angles  are  evidently  vertical 
angles. 

If  two  objects  are  in  the  same  horizontal  plane  with  the 
point  of  observation,  the  angular  distance  of  one  object  from 
the  other  is  called  its  bearing  from  that  object. 

If  two  objects  are  not  in  the  same  horizontal  plane  with 
cither  each  other  or  the  point  of  observation,  we  may  suppose 
vertical  lines  to  be  passed  through  the  two  objects,  and  to 
meet  the  horizontal  plane  of  the  point  of  observation  in  two 
points.  The  angular  distance  of  these  two  points  is  the  bear- 
ing of  either  of  the  objects  from  the  other.  It  may  also  be 
called  the  horizontal  distance  of  one  object  from  the  other. 


Note.  "Problems  in  Plane  Trigonometry"  can  be  obtained  in  pam- 
phlet form  of  Charles  W.  Sever,  Cambridge,  Mass. 


72  TRIGONOMETRY. 


Right  Triangles. 

2.  The  angle  of  elevation  of  a  tower  is  48°  19'  14",  and  the 
distance  of  its  base  from  the  point  of  observation  is  95  ft.  Find 
the  height  of  the  tower,  and  the  distance  of  its  top  from  the 
point  of  observation. 

3.  From  a  mountain  1000  ft.  high,  the  angle  of  depression 
of  a  ship  is  77°  35'  11".  Find  the  distance  of  the  ship  from 
the  summit  of  the  mountain. 

4.  A  flag-staff  90  ft.  high,  on  a  horizontal  plane,  casts  a 
shadow  of  117  ft.     Find  the  altitude  of  the  sun. 

5.  When  the  moon  is  setting  at  any  place,  the  angle  at  the 
moon  subtended  by  the  earth's  radius  passing  through  that 
place  is  57'  3".  If  the  earth's  radius  is  3956.2  miles,  what  is 
the  moon's  distance  from  the  earth's  centre? 

6.  The  angle  at  the  earth's  centre  subtended  by  the  sun's 
radius  is  16'  2",  and  the  sun's  distance  is  92,400,000  miles. 
Find  the  sun's  diameter  in  miles. 

7.  The  latitude  of  Cambridge,  Mass.,  is  42°  22'  49".  What 
is  the  length  of  the  radius  of  that  parallel  of  latitude  ? 

8.  At  what  latitude  is  the  circumference  of  the  parallel 
half  of  that  of  the  equator  ? 

9.  In  a  circle  with  a  radius  of  6.7  is  inscribed  a  regular 
polygon  of  thirteen  sides.     Find  the  length  of  one  of  its  sides. 

10.  A  regular  heptagon,  one  side  of  which  is  5.73,  is  in- 
scribed in  a  circle.     Find  the  radius  of  the  circle. 

11.  A  tower  93.97  ft.  high  is  situated  on  the  bank  of  a 
river.  The  angle  of  depression  of  an  object  on  the  opposite 
bank  is  25°  12'  54".     Find  the  breadth  of  the  river. 


MISCELLANEOUS    PROBLEMS.  73 

12.  From  a  tower  58  ft.  high  the  angles  of  depression  of 
two  objects  situated  in  the  same  horizontal  line  with  the  base 
of  the  tower,  and  on  the  same  side,  are  30°  13'  18''  and  45° 
46'  14".     Find  the  distance  between  these  two  objects. 

13.  Standing  directly  in  front  of  one  corner  of  a  flat-roofed 

house,  which  is  150  ft.  in  length,  I  observe  that  the  horizontal 

angle  which  the  length  subtends  has  for  its  cosine  VJ,  and 

that  the  vertical  angle  subtended  by  its  height  has  for  its  sine 
3 
— -     What  is  the  height  of  the  house? 


V34 

14.  A  regular  pyramid,  with  a  square  base,  has  an  edge 
150  ft.  in  length,  and  the  length  of  a  side  of  its  base  is  200  ft. 
Find  the  inclination  of  the  face  of  the  pyramid  to  the  base. 

15.  From  one  edge  of  a  ditch  36  ft.  wide,  the  angle  of  ele- 
vation of  a  wall  on  the  opposite  edge  is  62°  39'  10".  Find  the 
length  of  a  ladder  which  will  reach  from  the  point  of  observa- 
tion to  the  top  of  the  wall. 

16.  The  top  of  a  flag-staff  has  been  broken  off,  and  touches 
the  ground  at  a  distance  of  15  ft.  from  the  foot  of  the  staff. 
The  length  of  the  broken  part  being  39  ft.,  find  the  whole 
length  of  the  staff. 

17.  From  a  balloon,  which  is  directly  above  one  town,  is 
observed  the  angle  of  depression  of  another  town,  10°  14'  9". 
The  towns  being  8  miles  apart,  find  the  height  of  the  balloon. 

18.  From  the  top  of  a  mountain  3  miles  high  the  angle  of 
depression  of  the  most  distant  object  which  is  visible  on  the 
earth's  surface  is  found  to  be  2°  13'  50".  Find  the  diameter 
of  the  earth. 

19.  A  ladder  40  ft.  long  reaches  a  window  33  ft.  high,  on 
one  side  of  a  street.  Being  turned  over  upon  its  foot,  it 
reaches  another  window  21  ft.  high,  on  the  opposite  side  of 
the  street.     Find  the  width  of  the  street. 


74  TRIGONOMETRY. 


20.  The  height  of  a  house  subtends  a  right  angle  at  a  win- 
dow on  the  other  side  of  the  street ;  and  the  elevation  of  the 
top  of  the  house,  from  the  same  point,  is  60°.  The  street  is 
30  ft.  wide.     How  high  is  the  house  ? 

21.  A  lighthouse  54  ft.  high  is  situated  on  a  rock.  The 
elevation  of  the  top  of  the  lighthouse,  as  observed  from  a  ship, 
is  4°  52',  and  the  elevation  of  the  top  of  the  rock  is  4°  2'. 
Find  the  height  of  the  rock,  and  its  distance  from  the  ship. 

22.  A  man  in  a  balloon  observes  the  angle  of  depression  of 
an  object  on  the  ground,  bearing  south,  to  be  35°  30f;  the 
balloon  drifts  2|  miles  east  at  the  same  height,  when  the  angle 
of  depression  of  the  same  object  is  23°  14'.  Find  the  height 
of  the  balloon. 

23.  A  man  standing  south  of  a  tower,  on  the  same  horizon- 
tal plane,  observes  its  elevation  to  be  54°  16' :  he  goes  east 
100  yds.,  and  then  finds  its  elevation  is  50°  8'.  Find  the 
height  of  the  tower. 

24.  The  elevation  of  a  tower  at  a  place  A  south  of  it  is  30° ; 
and  at  a  place  B,  west  of  A,  and  at  a  distance  of  a  from  it,  the 
the  elevation  is  18°.     Show  that  the  height  of  the  tower  is 

a  ;  the  tangent  of  18°  being         ^  ~  l     - 

V(2+2V5)  V(io  +  2V5) 

25.  A  pole  is  fixed  on  the  top  of  a  mound,  and  the  angles 
of  elevation  of  the  top  and  the  bottom  of  the  pole  are  60°  and 
30°.  Prove  that  the  length  of  the  pole  is  twice  the  height  of 
the  mound. 

26.  At  a  distance  (a)  from  the  foot  of  a  tower,  the  angle 
of  elevation  (A)  of  the  top  of  the  tower  is  the  complement  of 
the  angle  of  elevation  of  a  flag-staff  on  top  of  it.  Show  that 
the  length  of  the  staff  is  2a  cot  2  A. 

27.  A  line  of  true  level  is  a  line  every  point  of  which  is 
equally  distant  from  the  centre  of  the  earth.     A  line  drawn 


MISCELLANEOUS    PROBLEMS.  75 

tangent  to  a  line  of  true  level  at  any  point  is  a  line  of  appar- 
ent level.  If  at  any  point  both  these  lines  be  drawn,  and 
extended  one  mile,  find  the  distance  they  are  then  apart. 

28.  In  Problem  2,  determine  the  effect  upon  the  computed 
height  of  the  tower,  of  an  error  in  either  the  angle  of  eleva- 
tion or  the  measured  distance. 

Oblique  Triangles. 

29.  To  determine  the  height  of  an  inaccessible  object  situ- 
ated on  a  horizontal  plane,  by  observing  its  angles  of  elevation 
at  two  points  in  the  same  line  with  its  base,  and  measuring 
the  distance  of  these  two  points. 

30.  The  angle  of  elevation  of  an  inaccessible  tower,  situated 
on  a  horizontal  plane,  is  63°  26' ;  at  a  point  500  ft.  farther 
from  the  base  of  the  tower  the  elevation  of  its  top  is  32°  14'. 
Find  the  height  of  the  tower. 

31.  A  tower  is  situated  on  the  bank  of  a  river.  From  the 
opposite  bank  the  angle  of  elevation  of  the  tower  is  60°  13', 
and  from  a  point  40  ft.  more  distant  the  elevation  is  50°  19'. 
Find  the  breadth  of  the  river. 

32.  A  ship  sailing  north  sees  two  lighthouses  8  miles  apart, 
in  a  line  due  west ;  after  an  hour's  sailing,  one  lighthouse  bears 
S.W.,  and  the  other  S.S.W.     Find  the  ship's  rate. 

33.  To  determine  the  height  of  an  accessible  object  situated 
on  an  inclined  plane. 

34.  At  a  distance  of  40  ft.  from  the  foot  of  a  tower  on  an 
inclined  plane,  the  tower  subtends  an  angle  of  41°  19' ;  at  a 
point  60  ft.  farther  away,  the  angle  subtended  by  the  tower  is 
23°  45'.     Find  the  height  of  the  tower. 

35.  A  tower  makes  an  angle  of  113°  12'  with  the  inclined 
plane  on  which  it  stands ;  and  at  a  distance  of  89  ft.  from  its 
base,  measured  down  the  plane,  the  angle  subtended  by  the 
tower  is  23°  27'.     Find  the  height  of  the  tower. 


76  TRIGONOMETRY. 


36.  From  the  top  of  a  house  42  ft.  high,  the  angle  of  eleva- 
tion of  the  top  of  a  pole  is  14°  13' ;  at  the  bottom  of  the  house 
it  is  23°  19'.     Find  the  height  of  the  pole. 

37.  The  sides  of  a  triangle  are  17,  21,  28  ;  prove  that  the 
length  of  a  line  bisecting  the  greatest  side  and  drawn  from 
the  opposite  angle  is  13. 

38.  A  privateer,  10  miles  S.W.  of  a  harbor,  sees  a  ship  sail 
from  it  in  a  direction  S.  80°  E.,  at  a  rate  of  9  miles  an  hour. 
In  what  direction,  and  at  what  rate,  must  the  privateer  sail 
in  order  to  come  up  with  the  ship  in  1-J-  hours  ? 

39.  A  person  goes  70  yds.  up  a  slope  of  1  in  3J  from  the 
edge  of  a  river,  and  observes  the  angle  of  depression  of  an  ob- 
ject on  the  opposite  shore  to  be  2J°.  Find  the  breadth  of  the 
river. 

40.  The  length  of  a  lake  subtends,  at  a  certain  point,  an 
angle  of  46°  24',  and  the  distances  from  this  point  to  the  two 
extremities  of  the  lake  are  346  and  290  ft.  Find  the  length 
of  the  lake. 

41.  Two  ships  are  a  mile  apart.  The  angular  distance  of 
the  first  ship  from  a  fort  on  shore,  as  observed  from  the  second 
ship,  is  35°  14'  10"  ;  the  angular  distance  of  the  second  ship 
from  the  fort,  observed  from  the  first  ship,  is  42°  11'  53".  Find 
the  distance  in  feet  from  each  ship  to  the  fort. 

42.  Along  the  bank  of  a  river  is  drawn  a  base  line  of  500 
feet.  The  angular  distance  of  one  end  of  this  line  from  an 
object  on  the  opposite  side  of  the  river,  as  observed  from  the 
other  end  of  the  line,  is  53° ;  that  of  the  second  extremity 
from  the  same  object,  observed  at  the  first,  is  79°  12'.  Find 
the  perpendicular  breadth  of  the  river. 

43.  A  vertical  tower  stands  on  a  declivity  inclined  15°  to 
the  horizon.  A  man  ascends  the  declivity  80  ft.  from  the  base 
of  the  tower,  and  finds  the  angle  then  subtended  by  the  tower 
to  be  30°.     Find  the  height  of  the  tower. 


MISCELLANEOUS    PROBLEMS.  77 

44.  The  angle  subtended  by  a  tower  on  an  inclined  plane 
is,  at  a  certain  point,  42°  17';  325  ft.  farther  down,  it  is  21° 
47'.  The  inclination  of  the  plane  is  8°  53'.  Find  the  height 
of  the  tower. 

45.  A  cape  bears  north  by  east,  as  seen  from  a  ship.  The 
ship  sails  northwest  30  miles,  and  then  the  cape  bears  east. 
How  far  is  it  from  the  second  point  of  observation  ? 

46.  Two  observers,  stationed  on  opposite  sides  of  a  cloud, 
observe  its  angles  of  elevation  to  be  44°  56'  and  36°  4'.  Their 
distance  from  each  other  is  700  ft.  What  is  the  linear  height 
of  the  cloud  ? 

47.  From  a  point  B  at  the  foot  of  a  mountain,  the  eleva- 
tion of  the  top  A  is  60°.  After  ascending  the  mountain  one 
mile,  at  an  inclination  of  30°  to  the  horizon,  and  reaching  a 
point  C,  the  angle  ACB  is  found  to  be  135°.  Find  the  height 
of  the  mountain  in  feet. 

48.  From  a  ship  two  rocks  are  seen  in  the  same  right  line 
with  the  ship,  bearing  N.  15°  E.  After  the  ship  has  sailed 
northwest  5  miles,  the  first  rock  bears  east,  and  the  second 
northeast.     Find  the  distance  between  the  rocks. 

49.  From  a  window  on  a  level  with  the  bottom  of  a  steeple 
the  elevation  of  the  steeple  is  40°,  and  from  a  second  window 
18  ft.  higher  the  elevation  is  37°  30'.  Find  the  height  of  the 
steeple. 

50.  To  determine  the  distance  between  two  inaccessible 
objects  by  observing  angles  at  the  extremities  of  a  line  of 
known  length. 

51.  Wishing  to  determine  the  distance  between  a  church  A 
and  a  tower  B,  on  the  opposite  side  of  a  river,  I  measure  a 
line  CD  along  the  river  (C  being  nearly  opposite  A),  and  ob- 
serve the  angles  A CB,  58°  20' ;  ACB,  95°  20' ;  ABB,  ^°  SO'; 
BBC,  98°  45'.     CB  is  600  ft.     What  is  the  distance  required  ? 


78  TRIGONOMETRY. 


52.  Wishing  to  find  the  height  of  a  summit  A,  I  measure 
a  horizontal  base  line  CD,  440  yds.  At  C,  the  elevation  of  A 
is  37°  18',  and  the  horizontal  angle  between  D  and  the  sum- 
mit is  76°  18' ;  at  D,  the  horizontal  angle  between  C  and  the 
summit  is  67°  14'.     Find  the  height. 

53.  A  balloon  is  observed  from  two  stations  3000  ft.  apart. 
At  the  first  station  the  horizontal  angle  of  the  balloon  and  the 
other  station  is  75°  25',  and  the  elevation  of  the  balloon  is  18°. 
The  horizontal  angle  of  the  first  station  and  the  balloon,  meas- 
ured at  the  second  station,  is  64°  30'.  Find  the  height  of  the 
balloon. 

54.  Two  forces,  one  of  410  pounds,  and  the  other  of  320 
pounds,  make  an  angle  of  51°  37'.  Find  the  intensity  and  the 
direction  of  their  resultant. 

55.  An  unknown  force,  combined  with  one  of  128  pounds, 
produces  a  resultant  of  200  pounds,  and  this  resultant  makes 
an  angle  of  18°  24'  with  the  known  force.  Find  the  intensity 
and  direction  of  the  unknown  force. 

56.  At  two  stations,  the  height  of  a  kite  subtends  the  same 
angle  (A).  The  angle  which  the  line  joining  one  station  and 
the  kite  subtends  at  the  other  station  is  B ;  and  the  distance 
between  the  two  stations  is  a.  Show  that  the  height  of  the 
kite  is  \a  sin  A  sec  B. 

57.  Two  towers  on  a  horizontal  plane  are  120  ft.  apart.  A 
person  standing  successively  at  their  bases  observes  that  the 
angular  elevation  of  one  is  double  that  of  the  other  ;  but,  when 
he  is  half-way  between  them,  the  elevations  are  complementary. 
Prove  that  the  heights  of  the  towers  are  90  and  40  ft. 

58.  To  find  the  distance  of  an  inaccessible  point  C  from 
either  of  two  points  A  and  B,  having  no  instruments  to  meas- 
ure angles.  Prolong  CA  to  a,  and  CB  to  b,  and  join  AB,  Ab, 
and  Ba.  Measure  AB,  500;  a  A,  100;  aB,  560;  bB,  100; 
and  Ab,  550. 


MISCELLANEOUS    PROBLEMS.  79 

59.  Two  inaccessible  points  A  and  B,  are  visible  from  B, 
but  no  other  point  can  be  found  whence  both  are  visible. 
Take  some  point  C,  whence  A  and  D  can  be  seen,  and  meas- 
ure CB  200  ft.  ;  ABC,  89°  ;  A  CD,  50°  30'.  Then  take  some 
point  E,  whence  D  and  B  are  visible,  and  measure  BE,  200 ; 
BDB,  54°  30' ;  BEB,  88°  30'.  At  D  measure  ABB,  72°  30'. 
Compute  the  distance  AB. 

60.  To  compute  the  horizontal  distance  between  two  inac- 
cessible points  A  and  B,  when  no  point  can  be  found  whence 
both  can  be  seen.  Take  two  points  C  and  D,  distant  200  yds. 
so  that  A  can  be  seen  from  C,  and  B  from  B.  From  C  meas- 
ure CF,  200  yds.  to  B,  whence  A  can  be  seen  ;  and  from  B 
measure  BE,  200  yds.  to  E,  whence  B  can  be  seen.  Measure 
AFC,  83°;  ACB,  53°  30' ;  ACF,  54°  31' ;  BBE,  54°  30'; 
BBC,  156°  25' ;  BEB,  88°  30'. 

61.  A  column  in  the  north  temperate  zone  is  east-southeast 
of  an  observer,  and  at  noon  the  extremity  of  its  shadow  is 
northeast  of  him.  The  shadow  is  80  ft.  in  length,  and  the 
elevation  of  the  column,  at  the  observer's  station,  is  45°.  Find 
the  height  of  the  column. 

62.  From  the  top  of  a  hill  the  angles  of  depression  of  two 
objects  situated  in  the  horizontal  plane  of  the  base  of  the  hill 
are  45°  and  30° ;  and  the  horizontal  angle  between  the  two 
objects  is  30°.  Show  that  the  height  of  the  hill  equals  the 
distance  between  the  objects. 

63.  Wishing  to  know  the  breadth  of  a  river  from  A  to  B, 
I  take  AC,  100  yds.  in  the  prolongation  of  BA,  and  then  take 
CB,  200  yds.  at  right  angles  to  AC.  The  angle  BBA  is  37° 
18'  30".     Find  AB. 

64.  The  sum  of  the  sides  of  a  triangle  is  100.  The  angle 
at  A  is  double  that  of  B,  and  the  angle  at  B  is  double  that 
at  C.     Determine  the  sides. 


80  TRIGONOMETRY. 


65.  If  sin7 A  +  5  cosM  =  3,  find  A. 

66.  If  sin2  A  =  m  cos  ^4  —  n,  find  cos ^4. 

67.  Given  sin  A  =  m  sin  B,  and  tan  A  —  n  tan  B,  find  sin  A 
and  cos  B. 

68.  If  tanM  +  4  sinM  =  6,  find  A. 

69.  If  sin  A  =  sin  2^4,  find  4. 

70.  If  tan  2  A  =  3  tan  A,  find  ^. 

71.  Prove  that  tan  50°  +  cot  50°  =  2  sec  10°. 

72.  Given  a  regular  polygon  of  n  sides,  and  calling  one  of 
them  a,  find  expressions  for  the  radii  of  the  inscribed  and  the 
circumscribed  circle  in  terms  of  n  and  a. 

If  P,  IT,  B  be  the  sides  of  a  regular  inscribed  pentagon, 
hexagon,  and  decagon,  prove  P2  =  H2  +  B2. 

Areas. 

73.  Obtain  the  formula  for  the  area  of  a  triangle,  given  two 
sides  b,  c,  and  the  included  angle  A. 

74.  Obtain  the  formula  for  the  area  of  a  triangle,  given  two 
angles  A  and  B,  and  included  side  c. 

75.  Obtain  the  formula  for  the  area  of  a  triangle,  given  the 
three  sides. 

76.  If  a  is  the  side  of  an  equilateral  triangle,  its  area  is 

4 

77.  Two  consecutive  sides  of  a  rectangle  are  52.25  ch.  and 
38.24  ch.     Find  its  area. 

78.  Two  sides  of  a  parallelogram  are  59.8  ch.  and  37.05  ch., 
and  the  included  angle  is  72°  10'.     Find  the  area. 

79.  Two  sides  of  a  parallelogram  are  15.36  ch.  and  11.46 
ch.,  and  the  included  angle  is  47°  30'.     Find  its  area. 


MISCELLANEOUS    PROBLEMS.  81 

80.  Two  sides  of  a  triangle  are  12.38  ch.  and  6.78  ch.,  and 
the  inclined  angle  is  46°  24'.     Find  the  area. 

81.  Two  sides  of  a  triangle  are  18.37  ch.  and  13.44  ch.,  and 
they  form  a  right  angle.     Find  the  area. 

82.  Two  angles  of  a  triangle  are  76°  54'  and  57°  33'  12", 
and  the  included  side  is  9  ch.     Find  the  area. 

83.  Two  sides  of  a  triangle  are  19.74  ch.  and  17.34  ch. 
The  first  bears  N.  82°  30'  W. ;  the  second,  S.  24°  15'  E.  Find 
the  area. 

84.  The  three  sides  of  a  triangle  are  49  ch.,  50.25  ch.,  and 
25.69  ch.     Find  the  area. 

85.  The  three  sides  of  a  triangle  are  10.64  ch.,  12.28  ch., 
and  9  ch.     Find  the  area. 

86.  The  sides  of  a  triangular  field,  of  which  the  area  is  14 
acres,  are  in  the  ratio  of  3,  5,  7.     Find  the  sides. 

87.  In  the  quadrilateral  ABCD  we  have  AD,  17.22  ch. ; 
AD,  7.45  ch. ;  CD,  14.10  ch.  ;  DC,  5.25  ch. ;  and  the  diago- 
nal AC,  15.04  ch.     Required  the  area. 

88.  The  diagonals  of  a  quadrilateral  are  a  and  b,  and  they 
intersect  at  an  angle  D.  The  area  of  the  quadrilateral  is 
\  ab  sin  D. 

89.  The  diagonals  of  a  quadrilateral  are  34  and  56,  inter- 
secting at  an  angle  of  67°.     Find  the  area. 

90.  The  diagonals  of  a  quadrilateral  are  75  and  49,  inter- 
secting at  an  angle  of  42°.     Find  the  area. 

91.  The  area  of  a  regular  polygon  of  n  sides,  of  which  one 

.    no*      ,180° 

is  a,  is  — ■  cot 

4  n 

92.  One  side  of  a  regular  pentagon  is  25.     Find  the  area. 

93.  One  side  of  a  regular  hexagon  is  32.     Find  the  area. 


82  TRIGONOMETRY. 


94.  One  side  of  a  regular  decagon  is  46.     Find  the  area. 

95.  Find  the  area  of  a  circle  whose  circumference  is  74  ft. 

96.  Find  the  area  of  a  circle  whose  radius  is  125  ft. 

97.  In  a  circle  with  a  diameter  of  125  ft.  find  the  area  of  a 
sector  with  an  arc  of  22°. 

98.  In  a  circle  with  a  radius  of  44  ft.  find  the  area  of  a 
sector  with  an  arc  of  25°. 

99.  In  a  circle  with  a  diameter  of  50  ft.  find  the  area  of  a 
segment  with  an  arc  of  280°. 

100.  Find  the  area  of  a  segment  (less  than  a  semicircle),  of 
which  the  chord  is  20,  and  the  distance  of  the  chord  from  the 
middle  point  of  the  smaller  arc  is  2. 

101.  If  r  is  the  radius  of  a  circle,  the  area  of  a  regular  cir- 

180° 


cumscribed  polygon  of  n  sides  is  nr2  tan 


n 


The  area  of  a  regular  inscribed  polygon  is  -  r2  sin 

2  n 

102.  If  a  is  a  side  of  a  regular  polygon  of  n  sides,  the  area 

of  the  inscribed  circle  is  — ■  cot2 

4  n 

The  area  of  the  circumscribed  circle  is  — -esc2 

4  n 

103.  The  area  of  a  regular  polygon  inscribed  in  a  circle  is 
to  that  of  the  circumscribed  polygon  of  the  same  number  of 
sides  as  3  to  4.     Find  the  number  of  sides. 

104.  The  area  of  a  regular  polygon  inscribed  in  a  circle  is 
a  geometric  mean  between  the  areas  of  an  inscribed  and  a  cir- 
cumscribed regular  polygon  of  half  the  number  of  sides. 

105.  The  area  of  a  circumscribed  regular  polygon  is  an  har- 
monic mean  between  the  areas  of  an  inscribed  regular  polygon 
of  the  same  number  of  sides,  and  of  a  circumscribed  regular 
polygon  of  half  that  number. 


MISCELLANEOUS    PROBLEMS.  83 

106.  The  perimeter  of  a  circumscribed  regular  triangle  is 
double  that  of  the  inscribed  regular  triangle. 

107.  The  square  described  about  a  circle  is  four-thirds  the 
inscribed  dodecagon. 

108.  Two  sides  of  a  triangle  are  3  and  12,  and  the  included 
angle  is  30°.  Find  the  hypotenuse  of  an  isosceles  right  tri- 
angle of  equal  area. 

Plane  Sailing. 

109.  Plane  Sailing  is  that  branch  of  Navigation  in  which 
the  surface  of  the  earth  is  considered  a  plane.  The  problems 
which  arise  are  therefore  solved  by  the  methods  of  Plane 
Trigonometry. 

The  following  definitions  will  explain  the  technical  terms 
which  are  employed  : 

The  difference  of  latitude  of  two  places  is  the  arc  of  a  merid- 
ian comprehended  between  the  parallels  of  latitude  passing 
through  those  places. 

The  departure  between  two  meridians  is  the  arc  of  a  par- 
allel of  latitude  comprehended  between  those  meridians.  It 
evidently  diminishes  as  the  distance  from  the  equator  at  which 
it  is  measured  increases. 

When  a  ship  sails  in  such  a  manner  as  to  cross  successive 
meridians  at  the  same  angle,  it  is  said  to  sail  on  a  rhumb-line. 
The  constant  angle  which  this  line  makes  with  the  meridians 
is  called  the  course,  and  the  distance  between  two  places  is 
measured  on  a  rhumb-line. 

If  we  neglect  the  curvature  of  the  earth,  and  consider  the 
distance,  departure,  and  difference  of  latitude  of  two  places  to 
be  straight  lines,  lying  in  one  plane,  they  will  form  a  right 
triangle,  called  the  triangle  of  plane  sailing.  If  ABB  be  a 
plane  triangle,  right-angled  at  D,  and  AB  represent  the  dif- 
ference of  latitude  of  A  and  B,  BAB  will  be  the  course  from 


84  TRIGONOMETRY. 


A  to  B,  AB  the  distance,  and  DB  the  departure,  measured 
from  B,  between  the  meridian  of  A  and  that  of  B. 

110.  Taking  the  earth's  equatorial  diameter  to  be  7925.6 
miles,  find  the  length  in  feet  of  the  arc  of  one  minute  of  a 
great  circle.* 

111.  A  ship  sails  from  latitude  43°  45'  S.,  on  a  course  N. 
by  E.,  2345  miles.  Find  the  latitude  reached,  and  the 
departure  made. 

112.  A  ship  sails  from  latitude  1°  45'  N.,  on  a  course  S.E. 
by  E.,  and  reaches  latitude  2°  31'  S.  Find  the  distance,  and 
the  departure. 

113.  A  ship  sails  from  latitude  13°  17'  EL,  on  a  course  N.E. 
by  E.  f  E.,  until  the  departure  is  207  miles.  Find  the  dis- 
tance, and  the  latitude  reached. 

114.  A  ship  sails  on  a  course  between  S.  and  E.,  244 
miles,  leaving  latitude  2°  52'  S.,  and  reaching  latitude  5°  8' 
S.     Find  the  course,  and  the  departure. 

115.  A  ship  sails  from  latitude  32°  18'  N.,  on  a  course  be- 
tween N.  and  W.,  making  a  distance  of  344  miles,  and  a 
departure  of  103  miles.  Find  the  course,  and  the  latitude 
reached. 

116.  A  ship  sails  on  a  course  between  S.  and  E.,  making 
a  difference  of  latitude  136  miles,  and  a  departure  203  miles. 
Find  the  distance,  and  the  course. 

117.  A  ship  sails  due  north  15  statute  miles  an  hour,  for 
one  day.  What  is  the  distance,  in  a  straight  line,  from  the 
point  left  to  the  point  reached  ?  (Take  earth's  radius,  3962.8 
statute  miles.) 

*  The  length  of  the  arc  of  one  minute  of  a  great  circle  of  the  earth 
is  called  a  geographical  mile,  or  a  knot.  In  the  following  problems,  this 
is  the  distance  meant  by  the  term  "  mile,"  unless  otherwise  stated. 


MISCELLANEOUS   PROBLEMS.  85 


Parallel  and  Middle  Latitude  Sailing. 

118.  The  difference  of  longitude  of  two  places  is  the  angle 
at  the  pole  made  by  the  meridians  of  these  two  places ;  or,  it 
is  the  arc  of  the  equator  comprehended  between  these  two 
meridians. 

119.  In  Parallel  Sailing,  a  vessel  is  supposed  to  sail  on  a 
parallel  of  latitude ;  that  is,  either  due  east  or  due  west. 
The  distance  sailed  is,  in  this  case,  evidently  the  departure 
made  ;  and  the  difference  of  longitude  made  depends  on  the 
solution  of  the  following  problem  : 

120.  Given  the  departure  between  any  two  meridians  at 
any  latitude,  find  the  angle  which  those  meridians  make, 
or  the  difference  of  longitude  of  any  point  on  one  meridian 
from  any  point  on  the  other.  (The  earth  is  considered  to 
be  a  perfect  sphere,  and  the  solution  depends  on  simple 
geometric  and  trigonometric  principles.  Cf.  Problem  7.) 
The  solution  gives  the  following  formula  : 

Diff.  long.  =  depart.  X  sec.  lat. 

121.  A  ship  in  latitude  42°  16'  N.,  longitude  72°  16'  W., 
sails  due  east  a  distance  of  149  miles.  What  is  the  position 
of  the  point  reached  ? 

122.  A  ship  in  latitude  44°  49'  S.,  longitude  119°  42'  E.,  sails 
due  west  until  it  reaches  longitude  117°  16'  E.  Find  the  dis- 
tance made. 

123.  In  Middle  Latitude  Sailing,  the  departure  between  two 
places,  not  on  the  same  parallel  of  latitude,  is  considered  to 
be,  approximately,  the  departure  between  the  meridians  of 
those  places,  measured  on  that  parallel  of  latitude  which  lies 
midway  between  the  parallels  of  the  two  places.  Except 
in  very  high  latitudes  or  excessive  runs,  such  an  assumption 
produces  no  great  error.  By  the  formula  of  Art.  120,  then, 
we  shall  have  — 

Diff.  long.  =  depart.  X  sec.  mid.  lat. 


86  TRIGONOMETRY. 


124.  A  ship  leaves  latitude  31°  14'  N.,  longitude  42°  19 
W.,  and  sails  E.N.E.  325  miles.     Find  the  position  reached. 

125.  Find  the  bearing  and  distance  of  Cape  Cod  from 
Havana.  (Cape  Cod,  42°  2'  N.,  70°  3'  W. ;  Havana,  23°  9'  N., 
82°  22'  W.) 

126.  Leaving  latitude  49°  57'  N.,  longitude  15°  16'.W.,  a 
ship  sails  between  S.  and  W.  till  the  departure  is  194  miles, 
and  the  latitude  is  47°  18'  N.  Find  the  course,  distance, 
and  longitude  reached. 

127.  Leaving  latitude  42°  30'  N.,  longitude  58°  51'  W.,  a 
ship  sails  S.E.  by  S.  300  miles.     Find  the  position  reached. 

128.  Leaving  latitude  49°  57'  N.,  longitude  30°  W.,  a  ship 
sails  S.  39°  W.,  and  reaches  latitude  47°  44'  N.  Find  the 
distance,  and  longitude  reached. 

129.  Leaving  latitude  37°  N.,  longitude  32°  16'  W.,  a  ship 
sails  between  N.  and  W.  300  miles,  and  reaches  latitude  41° 
N.     Find  the  course,  and  longitude  reached. 

130.  Leaving  latitude  50°  10'  S.,  longitude  30°  E.,  a  ship 
sails  E.S.E.,  making  160  miles'  departure".  Find  the  distance, 
and  position  reached. 

131.  Leaving  latitude  49°  30'  N.,  longitude  25°  W.,  a  ship 
sails  between  S.  and  E.  215  miles,  making  a  departure  of  167 
miles.     Find  the  course,  and  position  reached. 

132.  Leaving  latitude  43°  S.,  longitude  21°  W.,  a  ship  sails 
273  miles,  and  reaches  latitude  40°  17'  S.  "What  are  the  two 
courses  and  longitudes,  either  one  of  which  will  satisfy  the 
data? 

133.  Leaving  latitude  17°  N.,  longitude  119°  E.,  a  ship  sails 
219  miles,  making  a  departure  of  162  miles.  What  four  sets 
of  answers  do  we  get  ? 


MISCELLANEOUS    PROBLEMS.  87 

134.  A  ship  in  latitude  30°  sails  due  east  360  statute  miles. 
What  is  the  shortest  distance  from  the  point  left  to  the  point 
reached  ? 

Solve  the  same  problem  for  latitude  45°,  60°,  etc. 


Traverse  Sailing. 

135.  Traverse  Sailing  is  the  application  of  the  principles  of 
Plane  and  Middle  Latitude  Sailing  to  cases  when  the  ship 
sails  from  one  point  to  another  on  two  or  more  different 
courses.  Each  course  is  worked  up  by  itself,  and  these 
independent  results  are  combined,  as  may  be  seen  in  the 
solution  of  the  following  example  : 

136.  Leaving  latitude  37°  16'  S.,  longitude  18°  42'  W.,  a 
ship  sails  N.E.  104  miles,  then  N.N.W.  60  miles,  then  W.  by 
S.  216  miles.  Find  the  position  reached,  and  its  bearing  and 
distance  from  the  point  left. 

We  have,  for  the  first  course,  difference  of  latitude  73.5  N., 
departure  73.5  E. 

We  have,  for  the  second  course,  difference  of  latitude,  55.4 
N.,  departure  23  W. 

We  have,  for  the  third  course,  difference  of  latitude  42.1  S., 
departure  211.8  W. 

On  the  whole,  then,  the  ship  has  made  128.9  miles  of 
north  latitude,  and  42.1  miles  of  south  latitude.  The  place 
reached  is  therefore  on  a  parallel  of  latitude  86.8  miles  to 
the  north  of  the  parallel  left ;  that  is,  in  latitude  35°  49'.2  S. 

The  departure  is,  in  the  same  way,  found  to  be  161.3  miles 
W.  ;  and  the  middle  latitude  is  36°  32'.6.  With  these  data, 
and  the  formula  of  Art.  126,  we  find  the  difference  of  longi- 
tude to  be  201  miles,  or  3°  21'  W.  Hence  the  longitude 
reached  is  22°  3'  W. 

With  the  difference  of  latitude  86.8  miles,  and  the  depart- 
ure 161.3  miles,  we  find  the  course  to  be  N.  61°  43'  W.,  and 


X** 


88  TRIGONOMETRY. 


the  distance  183.2  miles.  The  ship  has  reached  the  same 
point  that  it  would  have  reached,  if  it  had  sailed  directly  on 
a  course  N.  61°  43'  W.,  for  a  distance  of  183.2  miles. 

137.  A  ship  leaves  Cape  Cod  (Ex.  125),  and  sails  S.E.  by 
S.  114  miles,  N.  by  E.  94  miles,  W.N.W.  42  miles.  Solve  as 
in  Art.  139. 

138.  A  ship  leaves  Cape  of  Good  Hope  (latitude  34°  22'  S., 
longitude  18°  30'  E.),  and  sails  N.W.  126  miles,  N.  by  E.  84 
miles,  W.S.W.  217  miles.     Solve  as  in  Ex.  136. 


*v 


fl 


EXAMINATION  PAPERS.  * 


£ 


V 


PLANE  TRIGONOMETRY. 

I. 
(Harvard  College,  Admission.    June,  1881.     Time,  H  hours.) 

1.  Define  a  logarithm.  What  is  the  logarithm  of  \  in  the 
system  of  which  16  is  the  base?  Find  the  logarithm  of  25  in 
the  same  system. 

2.  Compute  the  value  "^ff^^J  °7  logarithms. 

3.  Find  the  functions  of  127°  10'  from  your  trigonometric 
tables. 

4.  Prove  the  formula 

(cos  A  -  cos  Bf  +  (sin  A  -  sin  Bf  =--  4  sin2^=^?. 


5.  Two  sides  of  a  triangle  are  243  feet  and  188  feet,  and 
the  angle  opposite  the  second  side  is  42°  20'.  Solve  the  tri- 
angle completely. 

6.  A  pine  tree  growing  on  the  side  of  a  mountain,  which  is 
inclined  to  the  horizontal  at  an  angle  of  20°,  is  broken  by  the 
wind  but  not  severed  at  a  distance  of  40  feet  from  the  ground. 
The  top  falls  toward  the  foot  of  the  mountain,  and  strikes  the 
ground  50  feet  from  the  base  of  the  tree ;  find  the  height  of 
the  tree. 

*  Note.  In  these  papers,  as  in  many  text-books,  the  Greek  letters  a 
(alpha),  ft  (bayta),  y  (gamma),  6  (delta),  0  (thayta),  <p  (phee),  are  occasionally 
used  to  denote  angles. 


90  TRIGONOMETRY. 


II. 
{Harvard  College,  Admission.     June,  1SS2.      Time,  1\  hours.) 

1.  Explain  the  reason  of  the  rule  for  finding  the  character- 
istic (or  integral  part)  of  the  logarithm  of  a  number. 

Show  that  (according  to  this  rule)  the  mantissa  (or  frac- 
tional part)  is  always  positive. 

In  what  cases  is  the  logarithm,  as  a  whole,  positive,  and  in 
what  cases  negative  ? 

Thus,  state  clearly  the  value  of  the  logarithm  of  36,270  ;  of 
0.003627.  What  decimal  must  be  added  to  the  latter  loga- 
rithm to  produce  the  logarithm  of  0.01  ? 

2.  Find  the  time  required  to  increase  a  sum  of  money  a 
hundred  fold,  at  ten  per  cent  per  annum,  compound  interest, 
payable  yearly. 

3.  Find  the  formulas  for  the  trigonometric  functions  of 
90°+  a. 

4.  Find  by  the  tables  the  logarithms  of  the  trigonometric 
functions  of  290°  38'  (marking  the  signs). 

5.  An  observer  from  a  ship  saw  two  headlands.  The  first 
bore  E.N.E.  (i.e.  67°  30'  from  N.  towards  E.),  and  the  second 
N.W.  by  N.  (i.e.  33°  45'  from  N.  towards  W.).  After  he  had 
sailed  16.25  miles  N.  by  W.  (i.e.  11°  15'  from  N.  towards  W.), 
the  first  headland  bore  due  E.,  and  the  second  N.W.  by  W. 
(i.e.  56°  15'  from  N.  towards  W.).  Find  the  direction  and  dis- 
tance of  the  second  headland  from  the  first, 


6.    Prove  the  formulas 

cos  a  —  cos/? 
cosa-f  cos/3 

•    /)         2tan4-i 


tani(u  +  £)  tftn£(a  —  £), 


EXAMINATION    PArERS.  91 

III. 
{Harvard  College,  Freshman  Examination.    April,  1879.     Time,  3  hours.) 

1.  Prove  the  relations  between  the  sine,  cosine,  and  tan- 
gent of  90°-j-<£  and  the  functions  of  <f>.  Draw  a  figure  for  the 
case  where  </>  is  obtuse,  and  show  that  the  proof  still  holds 
good. 

Confirm  your  results  by  means  of  the  formulas  for  the  sine 
and  cosine  of  the  sum  of  the  two  angles. 

2.  Deduce  formulas  for  the  sine,  cosine,  and  tangent  of  2  a 
and  \  a,  in  terms  of  functions  of  a. 

3.  Prove  the  formula  : 

cos  (a  -f  /?)  sin/3  —  cos  (a  +  y)  siny= 
sin  (a  -f-  /5)  cos/3  —  sin  (a  -f-  y)  cosy. 

4.  Prove  that  in  any  triangle 

a2  =  b2  -f  c2  —  2  be  cos  A. 

5.  Deduce  the  formulas  for  the  tangents  of  the  half  angles 
of  a  triangle,  in  terms  of  the  sides. 

6.  Solve  the  triangles  : 

C=35°,  a  =  500,  c  =  250, 

B  =  22°22',       «  =  67.0G,        6  =  60.03. 

7.  The  Delta  measures  241  yds.  on  Cambridge  St.  and  115 
yds.  on  Quincy  St.,  and  the  angle  between  these  streets  is 
88°  52'.     Find  the  other  angles  of  the  Delta. 

8.  Find  the  area  of  the  Delta. 

9.  A  person  travelling  east  in  a  railroad  train  observes  a 
tower  situated  south  of  a  station  A,  and  on  the  same  horizontal 
plane  with  the  railroad.  At  the  station  B  he  finds  that  the 
distance  of  the  tower  is  2  miles ;  and  at  C,  3  miles  from  B,  its 
distance  is  4  miles.     Find  the  distance  of  the  tower  from  A. 


92  TRIGONOMETRY. 


IV. 

{Harvard  College,  Freshman  Examination.   April,  1880.     Time,  3  hours.) 

1.  Deduce  the  formula  sin  (a  +  ft)  = ,  drawing  the  figure 

for  the  case  in  which  a  is  in  the  second  quadrant  and  a  -f-  /3 
in  the  third. 

2.  Deduce  the  formulas  for  cos  2  a,  sin  *  a,  and  cos-|a,  in 
terms  of  functions  of  a. 

3.  Prove  the  theorem  of  the  sines. 

4.  From  the  fundamental  formulas  deduce  the  formula 

ta,n^(A  +  B)_a  +  b 
tea±(A  —  £)     a  —  b 

5.  Prove  that 

cos  (a  -j-  p  -t-  y)  _  ^  _  tan  a  tan  ^g  _  ^an  ^  tan    _  tan     ^_an  a 
cos  a  cos/j  cosy 

6.  In  a  triangle  B  =  4°  13.4'  and  a  =  2001,  give  all  the 
solutions  in  the  following  cases : 

(1)  5  =  150, 

(2)  5  =  200, 
(8)    5  =  2001. 

7.  A,  B,  and  Care  the  corners  of  a  triangular  field.  A  is 
40  ft.  W.  of  B  and  400  ft.  8.W.  of  C.  What  is  the  area  of 
the  field  ?    What  is  the  length  of  the  fence  which  encloses  it  ? 

8.  From  two  corners  of  the  Delta,  A  and  B,  lines  which 
make  angles  of  19°  52'  and  57°  32'  respectively  with  the  side 
AB  meet  directly  under  Memorial  Hall  tower.  The  length 
of  AB  is  345.1  ft.,  and  the  angle  of  elevation  of  the  tower  at 
A  is  32°  26'.  Find  the  height  of  the  tower,  and  its  angle  of 
elevation  at  B. 


EXAMINATION     PAPERS.  93 

V. 
{Harvard  College,  Freshman  Examination.    April,  1881.    Time,  3  hours.) 

1.  Deduce  the  formulas  which  connect  the  functions  of 
(90°  -f  4>)  and  <f>. 

2.  Prove  the  fundamental  formula  for  cos  (a  +  /J). 

3.  From  the  formula  just  found,  obtain  three  values  for 
cos  2  a. 

4.  Find  the  values  of  sin-  and  cos -,  in  terms  of  cos  a. 

5.  Prove  the  formula 

(cos  A  -  cos  B)2  +  (sin  A  -  sin  Bf  =  4  sin2  ^-=^- 

6.  Solve  the  following  triangles  : 

6  =  2434,  c  =  1881,  c  =  42°22\ 

a  =  0.00543,       c  =  0.07003,         a  =  4°  27'. 

7.  The  sides  of  a  triangle  are  715,  541,  and  368 ;  find  one 
angle  and  the  area. 

8.  The  height  of  Memorial  Hall  tower  is  190  feet.  From  its 
top  the  angles  of  depression  of  the  corners  of  the  Delta  which 
lie  on  Cambridge  St.  are  57°  44'  and  16°  59',  and  the  angle 
subtended  by  the  line  joining  these  corners  is  99°  30'.  Find 
the  length  of  the  Delta  on  Cambridge  St. 


VI. 

(Harvard  College,  Freshman  Examination.    April,  1882.     Time,  3  hours.) 

1.  Obtain  the  formulas  which  connect  the  sine,  cosine,  and 
tangent  of  (180°  +  <f>)  with  the  functions  of  <£. 

2.  Assuming  the  formulas  for  the  sine  and  cosine  of  the 
sum  of  two  angles,  prove  that 

(1)  tan  (a +  /?)=_ 

(2)  sin-^a=  V-J(l  —  cos  a). 


94  TRIGONOMETRY. 


3.  Find  all  the  values  of  x}  between  0°  and  360°,  which  will 
satisfy  the  equations 

(1)  tan  x  =  2  sin  2x, 

(2)  (sin  x  +  cos  x)2  =  2  sin  2  x. 

4.  The  length  of  each  side  of  a  regular  dodecagon  is  24 
feet ;  find  the  radius  of  the  inscribed  circle  and  the  area  of  the 
polygon. 

5.  In  a  certain  triangle,  a  =  20,  B  =  3°  24',  C=  85°  31'. 
Find  c  by  aid  of  the  table  containing  the  values  of  8  and  T. 

6.  Prove  the  Theorem  of  Sines,  and  solve  the  triangles 

(1)  6  =  468,  <?  =  327,  (7=34°  15'; 

(2)  a=  0.003641,     c  =  0.08091,    A  =    5° 20'. 

7.  Given  a2  =  b2  -f  c2  —  2  be  cos  A ;   obtain  the  formula 


The  sides  of  a  triangle  are  a  =  2408,  5  =  2028,  c  =  1884; 
find  the  angle  A. 

8.  Hingham  is  12  miles  south-east  of  Boston  ;  Quincy  is  5f 
miles  west  of  Hingham.     How  far  is  Quincy  from  Boston  ? 

9.  A  person  ascending  Memorial  Hall  tower  stops  to  rest 
at  a  window,  and  notices  that  the  angle  of  elevation  of  the 
vane  on  Appleton  Chapel  is  3°  34'.  When  he  reaches  the  top 
of  the  tower,  190  feet  above  the  ground,  he  finds  that  the 
height  of  the  Chapel  subtends  an  angle  of  13°  9'.  The  hori- 
zontal distance  between  the  two  towers  being  492  feet,  find 
the  height  of  Appleton  Chapel  and  the  distance  of  the  window 
above  the  ground. 


EXAMINATION    PAPERS.  95 

VII. 

{Cambridge,  Eng.     2nd  Previous  Exam.,  Dec.  7,  1870.      Time,  2\  hours.) 

1*   Assuming  that  the  angle  subtended  at  the  centre  of  any 
circle  by  an  arc  equal  to  its  radius  is  a  constant  angle,  show 

that  any  angle  may  be  expressed  by  the  fraction      ^°  ,  the 
constant  angle  being  taken  as  the  unit.  1  ac  ms 

Find  the  length  of  the  arc  subtended  by  an  angle  of  60°  in 
a  circle  whose  radius  is  3  feet. 

2.  The  sine  of  a  certain  angle  is  J  ;  find  the  other  trigono- 
metrical ratios  of  the  angle. 

3.  Trace  the  change  in  sign  and  magnitude  in  the  tangent 
of  an  angle,  as  the  angle  increases  from  0°  to  360°. 

4.  Find,  by  a  geometrical  construction,  the  cosine  of  G0° 
and  of  45°,  and  deduce  the  value  of  cos  3360°  and  cos  2565°. 

5.  Prove  the  formulas : 

(1)    sin  (A  —  B)  =  sin  A  cos  B  —  cos  A  sin  B, 
1  -  tanM 


(2)    cos  2  A 


1  +  teirM. 


/0x    sin  2  A  -f-  sin  4  A  0  A 

(3)    — — ■ — -  =  tan3A 

cos  2  J.  +  cos  4  A 

6.  Express  the  cosine  of  half  an  angle  in  terms  of  the  sine 
of  the  angle,  and  explain  the  double  sign. 

Employ  the  formula  to  find  the  value  of  cos  75°,  having 
given  sin  150°  =  -J-. 

7.  If  A,  B,  O  be  the  angles  of  a  triangle,  and  a,  b,  c  the 
sides  respectively  opposite  to  them,  show  that 


t    A         is  (s  —  a) 
cos-§-  A  =  +  \— ^ 


be 
where  s  —  one-half  the  sum  of  the  sides. 

8.    Find  the  greatest  angle  in  a  triangle  whose  sides  are  7 
feet,  8  feet,  and  9  feet. 

*For  aid  in  solving  this  and  similar  questions,  see  Wentworth  &  Hill's 
Tables,  pages  xi  and  xii. 


96  TRIGONOMETRY. 


VIII. 
[Cambridge,  Eng.     2nd  Previous  Exam.,  Dec.  7,  1877.     Time,  2\  hours.) 

1.  Define  the  cosine,  cotangent,  and  cosecant  of  an  angle, 
and  prove  that  these  ratios  remain  unchanged  so  long  as  the 
angle  is  the  same. 

Find  the  value  of  these  three  ratios  for  an  angle  of  45°. 

2.  Prove  the  formulas : 


(1)  sin  A  =  Vl  —  cos2A, 

(2)  cos^L  = - 


Vl  +  tanM 
If  sec  A  =  V2,  find  tan  A.  Ans.  1. 

3.  Prove  that 

sin  (90°  +  A)  =  cos  A,  and  cos  (90°  +  A)  =  -  sin  A. 
Hence  show  that  cos  (180°  -J-  „)  =  —  cos  A. 

4.  Show  that  cos2  A  tan2  A  -f-  sin2  A  cot2  A  =  1. 

5.  Prove  that  cos  (A  -f-  B)  =  cos  A  cos  B  —  sin  A  sin  B. 
Hence  show  that 

^^t^±Q  =  CotAcotBcotC-cotA-cotB-cota 
sin  A  sin  B  sin  C 

6.  Given  that  sin^A  =  f ,  find  the  value  of  tan  A. 

Ans.  -2T4-. 

7.  Prove  that  the  sides  of  any  plane  triangle  are  propor- 
tional to  the  sines  of  the  angles  opposite  to  these  sides. 

If  2s  =  the  sum  of  the  three  sides  (a,  b,  c)  of  a  triangle,  and 
if  A  be  the  angle  opposite  to  the  side  a,  prove  that 


2 

sin  A  =  ^p/s  (s  —  a)(s  —  b)  (s  —  c). 

8.    Prove  that  in  any  plane  triangle 

tan^(^-i?)  =  ^^cotJ(7. 


EXAMINATION    PAPERS.  97 

9.  If  the  side  a  and  the  angles  A  and  B  of  a  triangle  be 
known,  prove  that  the  side  b  may  be  found  by  means  of  the 
formula, 

log  b  =  log  a  -f-  log  sin  B  —  log  sin  A. 

Find  b,  having  given  that  a  =  1000  yards,  ^  =  50°,  ^=64°. 

Arts.  1173.29  yards. 

10.  The  minute-hand  of  a  clock  is  3  feet  6  inches  in  length ; 
find  how  far  its  point  will  move  in  a  quarter  of  an  hour,  it 
being  assumed  that  -n-  =  *f-.  Ans.  5  feet  6  inches. 


IX. 

{Cambridge,  Eng.    2nd  Previous  Exam.,  Dec.  10,  1878.      Time,  2\  hours.) 

1.  Define  sine,  cotangent;  and  prove  that  sin2 A-\- cos2 A=l. 
Express  the  other  trigonometrical  ratios  in  terms  of  the  cosine. 

2.  What  is  meant  by  the  circular  measure  of  an  angle? 
How  is  the  number  of  degrees  in  an  angle  found  from  its  cir- 
cular measure  ?  How  many  degrees  are  in  the  unit  of  circular 
measure  ? 

3.  Prove  that 

(1)  sin  (180°-M)  --  sin  A, 

(2)  tan  (90°+^)  =  -cot A. 
What  is  the  use  of  these  equations  ? 

4.  Find  the  general  form  of  all  the  angles  whose  sine  is  the 
same  as  sin  0. 

Write  down  the  sines  of  all  the  angles  which  are  multiples 
of  30°  and  less  than  360°. 


TRIGONOMETRY. 


5.    Prove  the  following  relations : 

(1)  cos  ( A  ~  B)  =  cos  A  cos  B  -f-  sin  A  sin  B, 

(2)  sin^  +  sin^  =  2sin|-(^  +  ^)  coa$(A-B), 

(3)  tan2  J. 


1  —  cos  2  A 


1  -f  cos  2  ^4 

6.  Find  cos  30°,  tan  45°,  sin  15°. 

7.  If  tan  A  +  sec  A  =  2,  prove  that  sin  .4  =  f  when  ^4.  is  less 
than  90°. 

If  sin  A  =  f,  prove  that  tan  ^4  +  sec  A  =  3  when  .4  is  less 
than  90°. 

8.  Prove  that  cos  3  A  =  4  cos3  A  —  3  cos  A,  and  find  tan  3  A 
in  terms  of  tan  A. 

9.  In  a  triangle  A,B,  C,  whose  sides  are  respectively  a,  b,  c, 
prove  that 

a)  ijii-^fiQ 

/Qn     sin  ^4 sin  i? sin  (7 

^z  j -     — 

a  b  c 

10.  Solve  a  triangle,  having  given  two  sides  and  the  angle 
opposite  one  of  them. 

Find  A,  B,  b,  having  given  a  =  25,  c  =  24,  C=  65° 59'. 


X. 

(Cambridge,  Eng.     2nd  Previous  Exam.,  Dec.  10,  1879.     Time,  2\  hours.) 

1.  Define  1°.     Assuming  that  %f-  is  the  circular  measure  of 
two  right  angles,  express  the  angle  A°  in  circular  measure. 

2.  Define  the  sine,  secant,  and  cotangent  of  an  angle,  and 
express  any  two  of  these  ratios  in  terms  of  the  third. 

Find  the  trigonometrical  ratios  of  the  angle  whose  cosine 
is  f . 


EXAMINATION   PAPERS.  99 

3.  Prove  that 

(1)  cos  (180°  +  A)  =  cos  (180°  -A), 

(2)  tan  (90°  +  A)  =  cot  (180°  -A). 

4.  Express  the  cosine  of  the  difference  of  two  angles  in 
terms  of  the  sines  and  cosines  of  these  angles. 

Prove  that 

*  tan-1  x  +  tan-1  y  =  tan"1  — XiL. 
l-xy 

5.  Prove  the  formulas  : 

/i\  .  o        x4-y        x  —  y 

(1)     cos  x  -f  cosy  =  A  cos  — L-2  cos  — — i» 


(2)  sinJa?4-cos-J-a?  =  ±  Vl  +  sin#, 

(3)  sina:(2cosa;—  1)  =  2sin|-a7  cosf.r. 

6.  Trace  the  changes  in  sign  and  magnitude  of 

2sinfl-sin2(9 
2sin^  +  sin2(9' 

as  ^  changes  from  0  to  2  tt. 

7.  Express  the  cosine  of  any  angle  of  a  triangle  in  terms  of 
the  sides  of  the  triangle. 

If  the  angle  opposite  the  side  a  be  60°,  and  if  b,  c  be  the 
remaining  sides  of  the  triangle,  prove  that 

(a+b+c)  (b+c-a)  =*Zbc. 

8.  Solve  a  triangle,  having  given  the  three  sides. 
Given  A  =  36°,  B  =  72°,  and  a  =  1 ;  solve  the  triangle. 

9.  The  sides  of  a  triangle  are  2,  3,  4;  find  the  least  angle. 

*  tan~1x=  arc  whose  tangent  is  x. 


100  TRIGONOMETRY. 


XL 

{Cambridge,  Eng.    2nd  Previous  Exam.,  Dec.  10,  1880.      Time,  2\  hours.) 

1.  Express  in  degrees,  minutes,  etc.,  (i.)  the  angle  whose 
circular  measure  is  -£$-*;  (ii.)  the  angle  whose  circular  measure 
is  5. 

If  the  angle  subtended  at  the  centre  of  a  circle  by  the  side 
of  a  regular  pentagon  be  the  unit  of  angular  measurement,  by 
what  number  is  a  right  angle  represented? 

2.  Find,  by  geometrical  constructions,  the  cosine  of  45°  and 
the  sine  of  120°. 

Prove  that 

(sin  30°  -f  cos  30°)  (sin  120°  -f  cos  120°)  =  sin  30°. 

3.  If  esc  -4  =  9,  find  cot  A  and  sec  A . 

4.  Prove  that 

cos(180°-M)  =  -cosA 
Find  the  value  of  (i.)  cot  840°;  (ii.)  sec37r. 

5.  Assuming  the  formula  for  the  sine  of  the  sum  of  two 
angles  in  terms  of  the  sines  and  cosines  of  the  separate  angles, 
find  (i.)  sin  75°  ;   (ii.)  sin  3  A  in  terms  of  sin  A. 

G.    Prove  the  formulas  : 

(1)  cos2  (A-B)  -sin2  (A  +  JB)  =  cos  2  A  cos  2B, 

(2)  1  -}-tan#  tan-J-#  =  sec#. 

7.    Prove  that 

cos^4+cos^  =  2cosJ(^4  +  ^)  cos|(4— B), 


-,  cos  5  0  -f-  cos  0  •     i 

and  express  — — as  a  single  term. 

r        cos50-cos<9  5 

8.    Solve  the  equations  : 

(1)  5tan2#  +  sec2#  =  tjt 

(2)  cos50  +  cos30  =  V2  cos40. 


EXAMINATION    PAPERS.  101 

9.  Prove  that  in  any  triangle  cos  .4  =  — -1— 

2  be 

Obtain  the  formula  for  tan  \A  in  terms  of  the  sides. 

10.  Find  an  expression  for  the  area  of  a  triangle  in  terms 
of  its  sides.  The  lengths  of  the  sides  of  a  triangle  are  3  feet, 
5  feet,  and  6  feet ;  what  is  its  area  ? 

11.  Given  that 

sin  38° 25'  =  0.6213757,    sin  38°  26'  =  0.6216036; 
find  the  angle  whose  sine  is  0.6215000. 


XII. 

{Cambridge,  Eng.      2nd  Previous  Exam.,  Dec.  10,  1881.     Time,  2\  hours.) 

1.  Define  the  unit  of  circular  measure.  The  ratio  of  the 
circumference  of  a  circle  to  its  diameter  being  3.14159,  find 
the  circular  measure  of  an  angle  of  126°. 

2.  Define  the  tangent,  cotangent,  and  cosecant  of  an  angle. 
Find  the  tangent  and  cotangent  of  an  angle  whose  cosecant  is 
1.25. 

3.  Trace  the  changes  in  sign  and  magnitude  of  sin  A  as  A 
changes  from  90°  to  270°. 

4.  Prove  the  following : 

(1)    tan(ir-f--4)  =  tan  A, 

tan  A  —  tan  B 


(2)    \&n(A-B) 

n  A  -f  sin  J 
>s  A  —  cos  B 


1  +  tan  A  tan  B 
(3)     sin^  +  sin^  =  cot^-^ 


102  TRIGONOMETRY. 


5.  Determine  the  value  of  cos  18°,  and  prove  that 

cos  36°  =  cos  60°  + cos  72°. 

6.  Show  that  for  certain  values  of  the  angles 

2cos£v4  =  Vl  +  sin^4.  —  Vl  —sin  A. 

Is  this  formula  true  for  values  of  A  lying  between  200°  and 
220°  ?  and  if  not,  how  must  it  be  modified  ? 

7.  Prove  that  in  any  triangle,  with  the  usual  notation, 

i  A         ls(s  —  a) 

cos  £  A  =  \ l-±-- li 

\      be 

and  that  the  area  is  equal  to 

Vs  (s  —  a)  (s  —  b)(s  —  c). 
Show,  also,  that 

ain2 A  =  cos2B  -f  cos2C-f  2  cos  A  cos  B  cos  C. 

8.  When  one  side  of  a  triangle  and  the  two  adjacent  angles 
are  given,  show  how  to  solve  the  triangle. 

Find  the  greatest  side  of  the  triangle,  of  which  one  side  is 
2183  feet,  and  the  adjacent  angles  are  78°  14'  and  71°  24'. 


APPENDIX. 


FORMULAS. 

1.     sin2 A  -f  cosM  =  1. 


2.  tan.i=£^. 

cos  A 

(  sin  A  X  esc  ^4  =  1. 

3.  -j  cos  A  x  sec i*L 
I  tan  A  X  cot  A  =  1. 


r  §5. 


4.  sin  (x  -J-  3/)  =  sin  x  cosy  +  cos x  sin y.  1 

5.  cos  (#  -f-  y)  —  cos  x  cos  y  —  sin  x  sin  y. 


6.  tan(*  +  y)=J«»£+t«2L 

1  —  tan  a:  tan  y 

7.  cot(g  +  y)  =  cota;coty-1. 

cot  x  -f-  cot  y 

8.  sin  (x  —y)  =  sin  a;  cos  y  —  cos  a;  sin  y. 

9.  cos  (x  —y)  =  cos a;  cosy  -j-  sin  a;  sin  y. 

in      ,      f          s        tana;  —  tan  y 
10.     tan  (x  —  y)  =  = £-. 

1  -f-  tan  a:  tan  y 

n.  /            x       cot  x  cot  v  4-  1 
.     cot  (x  —  y)  — £— ! — 

cot  y  —  cot  a: 


'  §  3i. 


§32. 


12.     sin  2x  =  2  sin  a;  cos  a;. 


13.     cos  2  a;  =  cos2a;  —  sin2a;. 


§33. 


154 


FORMULAS. 


14. 
15. 
16. 
17. 


tan  2  x  = 
cot  2x  = 


2  tana: 
1  —  tan'2# 

cot2#  —  1 


2  cot  x 
sin  |  z  =  zb-y  — 

cos  I  z 


§33. 


cosz 


^ 


+  cosz 
2 


18.     tan  J  z 


:\i 


1  —  cos  z 


+  COS 


19. 
20. 
21. 
22. 
23. 

24. 

25. 

4  26. 

27. 

28. 


cot  hz 


-*aJE 


COS  z 


cosz 


§34. 


sin  A  -f  sin  B  =  2  sin  £  (^4  -f  i?)  cos  J  (^4  -  J5). 
sin  ^  -  sin  B  =  2  cos  \  {A  +  £)  sin  \(A-  B). 
cos  ^4  +  cosi?  -  2  cos  £  (^4  +  B)  cos  £  {A  —  B). 
cos  J.  —  cos2?  =  —  2  sin  i  (A-\-  B)  sin  £  (-4  —  B). 

sin  J.  +  sin  B  _  tan  £  (^4  -f  i?) 
sin  ^4  —  sin  B     tan  £  (^4  —  B) 


§35. 


§36. 


a sin  A 

b      sin  J9 

a2  =  ^  +  c2-2focosA     §37. 

a  —  b  =  tan  £  (^4  —  i?)     „  38 
a  +  b     toal(A  +  £)    S      ' 


8m  1^  =>-*)/«"«).    §43. 
X  be 


FORMULAS. 


155 


29. 
30. 


Is  (s  —  a) 


tan  ».!=>- *><'-*> 
\      s(s  —  a) 


31. 
32. 
33. 

34. 
35. 
36. 

37. 
38. 


l(s-a)(S-b)(s-c)_ 
r 


tan  2  A  = 

s  —  a 

F=  \  ac  sin  B. 


^§43. 


F=\/s(s-a)(s-b)(s-Cy 


tp     abc 

F=  h  r  (a  -f  b  -f-  c)  =  rs. 

Spherical  Trigonometry. 
cos  c  =  cos  a  cos  b. 

sin  a  =  sine  sin  A. 
sin  6  =  sin  c  sin  J9. 


§44. 


§46. 


40. 

41. 

42. 

43. 


cos  A  =  tan  6  cot  c. 
cos  j5  =  tan  a  cot  <?. 

cos  A  =  cos  a  sin  B. 
cos  i?  =  cos  b  sin  ^4. 

sin  &  =  tan  a  cot  ^4. 
sin  a  =  tan  5  cot  B. 


cos  c  =  cot  A  cot  -S. 

sin  asm  B  =  sin  5 sin  ^4.  ") 

sin  a  sin  (7  =  sin  <?  sin  A.  [  §  51. 

sin  bsin  C  =  sin  c  sin  B.  ) 


156 


FORMULAS. 


44. 


cos  a  =  cos  b  cos  c  -f-  sin  b  sin  <?  cos  A. 
cos  6  =  cos  a  cos  e  -f  sin  a  sin  c  cos  B. 
cos  c  =  cos  a  cos  6  -J-  sin  a  sin  6  cos  (7. 


cos  A  =  —  cos  .5  cos  C  -j-  sin  B  sin  C  cos  a. 

45.  \  cos  J3  =  —  cos  ^4  cos  C  +  sin  J[  sin  C  cos  6. 

cos  C  =  —  cos  .4  cos  i?  -f-  sin  ^4  sin  B  cos  <?. 


§51. 


46. 


sin  $  J.  —  Vsln  (s  —  b)  sin  (s  —  c)  esc  5  esc  c. 

cos  %  A  —  Vsin  s  sin  (s  —  a)  esc  6  esc  <?. 

tan  £  A  =  Vcsc  s  esc  (s  —  a)  sin  (s  —  6)  sin  (s  —  c). 


47. 


sin  $  a  —V—  cos  #  cos  (#  —  -4)  esc  -5  esc  C. 
cos  £  a  =  Vcos  (S  —  B)  cos  (#  —  C)  esc  i?  esc  C. 
tan  J  a  =V—  cosxS'cos  (JS—A)  sec  (S—B)  sec  (8—C). 


§52. 


r  cos  £  (^4  -f-  i?)  cos  i  c  =  cos  J  (a  -j-  6)  sin  J  C. 

4ft  J  sin  J  ( A  +  i?)  cos  I  c  =  cos  *  (a  —  5)  cos  $  C. 

1  cos  £  ( A  —  B)  sin  £  c  =  sin  $  (a  -f  b)  sin  £  C. 

'.  sin  $  ( J.  —  B)  sin  J  <?  =  sin  I  (a  —  b)  cos  J  (7. 


cos  i  (a  +  o) 


49. 


50. 


sm  j  (a  +  6) 

tznHa  +  b)    =£2liI4^|)tanJc. 

cos  *  (^4  -f-  B) 

tan*  (a -5)    =  s?n  \[AA  ~  ?>  tan  }  c. 
sin  £  (J.  -f  B) 

180° 


§53. 


51.     tan2  \  E=  tan  *  (s  —  a)  tan  £  (s  —  6)  tan  £  (s  —  c). 


§60. 


FORMULAS.  157 


Prof.  Blakslee's  construction  by  which  the  direction  ratios 
for  plane  right  triangles  give  directly  from  a  figure  the  analo- 
gies for  a  right  trihedral  or  for  a  right  spherical  triangle. 


The  construction  consists  of  two  parts. 

(a)  Lay  off  from  the  vertex  V&  unit's  distance  on  each  edge. 

(b)  Pass  through  the  three  extremities  of  these  distances 
three  planes  perpendicular  to  one  of  the  edges,  as  VA.  Now 
these  three  parallel  planes  will  cut  out  three  similar  right 
triangles.  The  first  being  constructed  in  either  of  the  two 
usual  ways,  the  construction  of  the  others  is  evident. 

Since  the  plane  angles  Aif  A2,  A3  all  equal  the  dihedral  A, 
and  the  nine  right  triangles  in  the  three  faces  give  the  values 
in  the  figure,  we  have  : 

(1)  sin  A  =  sin  a  :  sin  h ;  similarly,  sin  B  =  sin  b  :  sin  h. 

(2)  cos  A  =  tan b  :  tan  h ;  similarly,  cos  B  =  tana :  tan  h. 

(3)  tan  A  =  tana :  sin  b  ;  similarly,  tan  B  =  tan  b  :  sin  a. 

(4)  cos  h  =  cos  a    cos  b  ;  (by  3)  =  cot  A  cot  B. 

(5)  sin  A  =  cos  B :  cos  b ;  sin  B  =  cos  A  :  cos  a. 


Note.  If  a  sphere  of  unit  radius  be  described  about  Fas  a  centre,  the 
three  faces  will  cut  out  a  right  spherical  triangle,  having  the  sides  a,  b, 
and  h,  and  angles  A,  B,  and  H.  The  above  formulas  are  thus  seen  to 
be  the  analogies  of: 


158 


FORMULAS. 


(1)  sin  A  =  a  :  A  ;  sin  i?  =  Z>  :  A. 

(2)  cos  -4.  =  6  :  A ;  cos  5  =  a :  A. 

(3)  tan-4  =  a  ■  6  ;  tan  B  =  b :  a. 


(4)  A2 


a2  + 


1  =  sin2  +  cos2 ;  1  =  cot  A  cot  B. 


(5)  sin  A  =  cos  5  ;  sin  B  =  cos  A 

Napier's  rules  give  only  the  following,  which  follow  from  the  analo- 
gies as  numbered  : 


By  |  sin  a  =  sin  A  sin  A  =  tan  b  cot  B 
(1)  I  sin  6  =  sin  B  sin  A  =  tan  a  cot  ^1 


(5){ 


(3) 

(2) 


cos  A  =  sin  B  cos  a  —  tan  b  cot  A 

cos  5  =  sin  A  cos  &  =  tan  a  cot  A 

(4)  {  cos  A  =  cos  a  cos  b  =  cot  J.  cot  B  }  (4) 


The  Gauss  Equations. 


cos %(A  +  -5)  cos  | c  =  cos  J-(a  +  &)  sin  i  O. 
sin  £(.4  -f-  B)  cos  ^ c  =  cos  %{a,  —  b)  cos  £  (7. 
cos  \{A  —  B)  sin  J  c  =  sin  J  (a  -f-  5)  sin  £  C. 
sin  J(^L  -  5)  sin  \  c  =  sin  i(a  —  b)  cos  }  C. 
/iU±-B)    /**=    /i(a±i)    /la 

Rule  I.     sin  in  (I.)  gives  —  in  (3),  and  conversely, 
cos  in  (I.)  gives  -f  in  (3),  and  conversely. 

Rule  II.  Functions  have  same  names  in  (2)  and  (3). 
Functions  have  co-names  in  (4)  and  (1). 


ANSWEKS. 

PLANE    TRIGONOMETRY. 

Exercise  I. 

1.  sin£=-,  cos£  =  _,    tan£  =  _,    cotB  =  ®    secB  =  -,    cacB^Z 
c  c  a  o  a  b 

3.  (i.)  sin  =  §,  cos  =  $,        (ii.)  sin  =  A,  etc.        (v.)  sin  =  f f ,  etc. 

tan  =  f ,  cot  =  f ,       (iii.)  sin  =  T8T,  etc.       (vi.)  sin  —  |£$,  etc. 
sec  =  \ ,  esc  =  f.       (iv.)  sin  =  ?9T,  etc. 

4.  The  required  condition  is  that  a2  +  b2  =  c2.     It  is. 

5.  (i.)sin=-^ — -^,  etc.  (in.)  sin  =  -,  etc. 
v  '            m2  +  n2  v     '  s 

(ii.)  sin=  .  ^,,  etc.  (iv.)  sin  =  — ,  etc. 

7  x2  +  y2  qr 

7.  In  (iii.)p2^2  +  q2s2  =p2s2\  in  (iv.)  m2n2s2  +  m2p2  v2  =  n2q2r2. 

8.  c  =  145 ;        whence,  sin  A  =  T2¥%  =  cos  B ;  cos  J.  =  ££§  =  sin  i? ; 
tan  J.  =  T2¥4j  =  cot  i?  ;  cot  A  =  -\4^  «-  tan  B ;  sec  J.  =  Hf  =  esc  B  ;  eta 

9.  5  =  0.023  ;  whence,  tan  J.  =  cot  £  =  ^ ;  cot  A  -  tan  5  =  3ft*,  etc. 

10.  a  =  16.8 ;  whence,  sin  A  =  |f  §  =  cos  B,  etc. 

Vp2  +  g2 

11.  c  =  »  +  g;  whence,  sin  A  =-  — ±-  =  cosj5;  etc. 

12.  6  =  y/q  (p  +  q);  whence,  tan  A  =  -Vl-  =  cot -5 ;  etc. 

p  —  q 

13.  a  =  p  —  q:  whence,  sin  A  =  — ; —  =  cosi?;  etc. 

r      a  p  +  q 

14.  sin^4  =  fV£  =  0.89443;  etc.  15.   sin^4  =  f ;  etc. 

16.  sin  A  =  $  (5  +  Vl)  =  0.95572 ;  etc. 

17.  cos  A  =  £  (V3l-1)  =  0.57097;  sin  A  -|(VSl  +  l)- 0.82097;  etc. 

18.  a  =12.3.  20.   a  =  9.  22.   c  =  40. 

19.  6  =  1.54.  21.   6  =  68.  23.   c  =  229.62. 

24.   Construct  a  rt.  A  with  legs  equal  to  3  and  2  respectively;    then 
construct  a  similar  A  with  hypotenuse  equal  to  6. 
In  like  manner,  25,  26,  27,  may  be  solved. 
28.   a  -  1.5  miles  ;  b  =  2  miles.  31.   400,000  miles. 

30.  a  -  0.342.  6  =  0.940  ;  a  =  1.368,  6  =  3.760.       32.   142.926  yards. 


TRIGONOMETRY. 


Exercise  II. 

5.  Through  A  (Fig.  3)  draw  a  tangent,  and  take  AT=>3;  the  angle 

A OT  is  the  required  angle. 

6.  From  0  (Fig.  3)  as  a  centre,  with  a  radius  =  2,  describe  an  arc  cut- 

ting at  S  the  tangent  drawn  through  B ;   the  angle  SO  A  is  the 
required  angle. 

7.  In  Fig.  3,  take  OM=  %,  and  erect  MP  JL  OA  and  intersecting  the 

circumference  at  P;  the  angle  POM  is  the  required  angle. 

8.  Since  sin  x  =  cos  a;,  OM=  PM  (Fig.  3),  and  #  =  45°  ;  hence,  construct 

x  =  45°. 

9.  Construct  a  rt.  A  with  one  leg  =  twice  the  other ;  the  angle  opposite 

the  longer  leg  is  the  required  angle. 

10.  Divide  OA  (Fig.  3)  into  four  equal  parts ;  at  the  first  point  of  divi- 
sion from  0  erect  a  perpendicular  to  meet  the  circumference  at 
some  point  P.     Join  OP;  the  angle  A  OP  is  the  required  angle. 

21.   r  sin#.  22.    Leg  adjacent  to  A  =  nc,  leg  opposite  to  A  =  mc. 

Exercise  III. 

1.  cos  60°.  cotl°.  sec  71°  50'.  tan  7°  41'; 
sin  45°.               .  tan  75°.                sin  52°  36'.  sec  35°  14'. 

2.  cos  30°.  cot  33°.  sec  20°  58'.  tan0°  1'. 
sin  15°.                tan  6°.                  sin  4°  21'.                   sec  44°  59. 

3.  £V3. 

4.  tan  4=  cot  4  =  cot  (90° -4);  hence,  A  =  90°-  A  and  4  =  45°. 

5.  30°.  7.   90°.  9.   22°  30'.  11.    10°. 

6.  30°  8.   60°.  10.    18°.  12      90° 

n  +  1 

Exercise  V. 

1 .  cos  A  =  y\,  tan  A  =  ^-,        cot  A  =  T\,    sec  A=  ^-,        esc  A  =  -j-f. 

2.  cos  A  =  0.6,  tan -4  =  1.3333,  cot  4  =  0.75,  sec4  =  1.6667,  esc 4  =  1.25. 

3.  sin  A  =  -H-,  tan  A  -  ££,        cot  4  =  f  f ,    sec  A  =  f  £,        esc  4  =  f  f 

4.  sin  A  =  0.96,  tan  J.  =  3.42854,  cot  A  =  0.29167,  sec  A  =  3.5714. 

5.  sin  A  =  0.8,  cos  A  -  0.6,  cot  A  -  0.75,  sec  4  -  1.6667,  esc  4  =  1.25. 

6.  sinA  =  £V2,  cos4=jV2,  tanA=l,       secA=\/2,    csc4  =  \/2.~ 

7.  tan  4  =  2,        sin  4  =0.90,    cos  4  =  0.45,  sec  A  =  2.22,  esc  A  -1.11. 


ANSWERS. 


8.  cos  A  =  £,  sin  A  =  $V%,  tanA=V3,  cot  A=^Vs,  cscJ.  =  §V3. 

9.  sin  A  =  I  V%  cos  A  =  I V2,  tan  J.  =  1,  cot  A  =  1,  sec  .A  —  V2. 

10.  cos  4=  VI -ra2,  tan4  =  — m—  Vl-m2,     cot  A  =  - y/TZZrf 

1  —  m2  m 

ii  ^      1— m2  ,<         2ra  ,   A      1— m2  .      1+m2 

11.  cosA=- — ^-i      tan  A= -•    cot -4= »    sec  A  =    T    - 


l+m2  1-m2  2  m 

1 0      •      j,      m2  —  n2  >i      m2  —  n2  A      m2  +  n2 

12.  sin  ^4=- 1     tan4  =  — •  sec  A  =  -— : 

m?  +  n2  2mn  zmn 

13.  cot  =  l,  sin=£V2,    cos  =  £ V2,  sec  =  V2,       csc=V2. 

14.  cos  =  jV3,    tan  =  ^V3,    cot=V3,  sec  =  §V3,    esc  =  2. 

15.  sin  —  £  V3,    cos  =  J,  tan  =  V3,  cot  =  £ V3,    sec  =  2. 

16.  sin  =  ^V2-\/3,    cos  =  ^V2  +  V3,  cot-2+V£ 

17.  sin  =  |V2-V2,    cos  =  £  V2  +  V2,  tan=V2-l. 

18.  cos  =  1,  tan  =  0,  cot  =  oo,  sec  =  l,  esc  =  oo. 

19.  cos  =  0,  tan  =  oo,  cot  =  0,  sec  =  oo,  esc  =  1. 

20.  sin  =  1,  cos  =0,  cot  =  0,  sec  =  oo,  csc  =  l. 

21.  cosA=Vl-sinM,      tan  A  =  —  ™±A *  Csc4  =  -^L-. 

Vl-smM  sin^ 

22.  sin^=Vl-cos2vl,      tan  A  =  Vl  ~  C0S'M,    Cot4  = *»■*  . 

,  C0*A  Vl-~co~sM 

sec  A  = -,  esc  A 


cosA  Vl-cosM* 

23.  sin^=       tan^  ^     cos^^  1       _,    cot^  =  -J_, 

VI  +  tan2  A  Vl  +  tan2J.  tan  .4 

sec  A  -  Vl+tanM,     esc  4  =  ^l±5i3. 

tan  ^4 

24.  tan  A  =  — — ,  Csc  A  -  Vl  +  cot2  4,    sin  4  = 


cot  ^4 


Vl  +  cot2  A 

VI  +  cot2  J.  cot  ^4 

25.  sin^  =  ^V5,     cos^=§V5.  27.   sin^  =  /T,    cosvl  =  ff 

26.  sin4  =  |Vl5,   tanJ.=Vl5.  2g    1  -  3cosM  +  3cosM 

cos2  A  —  cos4  A 


TRIGONOMETRY. 


Exercise 

VI. 

1. 

6.6                   5. 

-  =  cos  A ;  .".  c  =  j. 

c                             cos  A 

A  =  90°-B,    a  =  ccos£, 
6  =  c  sin  B. 

39.   c-  7.8112,  ^1  =  39°  48', 

B  =  50°  12'. 

40.   6=69.99,    -4  =  30' 12", 

B  =  89°  29'  48". 

41.   a  =1.1886,  ^1  =  43°  20', 

B  =  46°  40'. 

42.   6  =  21.249,    c-  22.372, 

5  =  71°  46'. 

43.   a  =6.6882,    c=  13.738, 

£  =  60°  52'. 

44.   a  =63.89,      6  =  23.369, 

5  =  20°    6'. 

45.  a-  19.40,      6  =  18.778, 

^4  =  45°  56'. 

46.   6  =  53.719,    c  =  71.377, 

A  =  41°  11'. 

47.  a  =12.981,    c  =  15.796, 

^1  =  55°  16'. 

48.   a  =0.58046,  6  =  8.442, 

A=    3°  56'. 

49. 

F=$(c2  sin  A  cos  A).                      51. 

^=£(6*  tan  4). 

50. 

^=i(a2cotA).                               52. 

.P  =  $(aVc2-a2). 

53. 

6  =  11.6,       c  -  15.315,  A  =  40°  45'  48",   B  =  49°  14'  12". 

54. 

a  =7.2,          c  =  8.766,     B  =34°  46'  40",   A  =  55°  13'  20". 

55. 

a  =  3.6474,   6  =  6.58,        c  =  7.5233, 

5=61°. 

56. 

a  =  10.283,   6  =  19.449.   A  =  27°  52' 

B  =  62°  8'. 

57. 

19°  28'  and  70°  32'.                      65. 

tan  A  =  £  ^1  =  59°  45'. 
6 

a  =  6  tan  ^4,    95.34. 

58. 

3  and  5.1961. 

bo. 
90° 

67. 


b  =  c  sin 


n+  1 
90°  68.    7.0712  miles  in  each  direction 


n  +  1  69.   20.88  feet. 

60.  36°  52' 12"  and  53°  7' 48".  70.   56.65  feet. 

61.  212.1  feet.  71.   228.63  yards. 

62.  732.22  feet.  72.   136.6  feet. 

63.  3270  feet.  73.  140  feet. 

64.  37.3  feet,  96  feet.  74.    84.74  feet. 

Exercise  VII. 

1.  C  =  2(  90°-^),  c  =  2acos^4,      A  =  asinA 

2.  A  =  5(180°  -C),  c  =  2acosA,      h-aunA. 

3.  C=2(90°-A),  a  =  c  +  2cosA,?i  =  asmA. 


ANSWERS. 


4.  A  =  J  (180°  -  C),  a  =  c  +  2cosA,  h-annA. 

5.  C  =  2  (90°  -A),  a  =  Ah-  sin  A,      c-=2acosA. 

6.  A  =  £(180°-C),  a  =  h  +  s'mA,      c  =  2acosA. 

7.  sin  A  =  A  H-  a,        C=  2(90°  -  A),  c  =  2  a  cos  A. 

8.  tanA=AH-ic,      C=2(90°~A),  a=A  +  sinA 

9.  A  =  67°  22'  50",     C=  45°  14'  20",  A  =  13.2. 

10.  e-  0.21943,  A  =  0.27384,       2?-  0.03004. 

11.  a  =  2.055,  A  =  1.6852,  i^=  1.9819. 

12.  a  =  7.705,  c  =  3.6676,         JP- 13.73, 

13.  A  -  79°  36'  30",    0-  20°  47',  c  =  2.4206. 

14.  A  =  77°  19'  11",    C=  25°  21'  38",  a  =  20.5. 

15.  A  =  25°  28',  C=  129°  4',  a  =  81.388,       A  =  35. 

16.  A  =  81°  12'  9",      C=  17°  35' 42",  a  =  17,  c  =  5.2 

17.  i^=icV4a2-c2.  22.  0.76537. 

18.  .F=a2sin£Ccos£C.  23.  94°  20'. 

19.  F=  a?  sin  A  cos  A.  24.  2.7261. 

20.  jF=A2tan£C.  25.  38°  56' 33". 

21.  28.284  feet,  4525.44  sq.  feet.  26.  37.7 

Exercise  VIII. 

1.  r  -  1.618,     A  =  1.5388,  i^=  7.694. 

2.  r  =  11.269,  A  =  10.886,  i^=  381.04. 

3.  A  =  0.9848,  p  =  6.2514,  F=  3.0782. 

4.  A  =  19.754,  c  =  6.2537,  F=  1236. 

5.  r  =  1.0824,  c  =0.8284,  F=  3.3137. 

6.  r  =  2.592,     A  =  2.488,      c=  1.4615. 

7.  r-  1.5994,  A  =  1.441,  p  =  9.716. 

8.  0.6181.  12.   0.2238.  17.   11.636. 

9.  0.64984.  13.   0.31.  18.   99.64. 
10.   0.51764.                      14.   0.82842.         19.   1.0235. 

c  15.   94.63.  20.   0.635. 


11.  6  = 


2cosy0_°  16.  415. 


TRIGONOMETRY. 


Exercise  IX. 

5.  Two  angles :  one  in  Quadrant  I.,  the  other  in  Quadrant  II. 

6.  Four  values:  two  in  Quadrant  L,  two  in  Quadrant  IV. 

7.  x  may  have  two  values  in  the  first  case,  and  one  value  in  each  of 

the  other  cases. 

8.  If  cos#  =  —  f,  x  is  between  90°  and  270°;  if  cot  a  =  4,  x  is  between 

0°  and  90°  or  180°  and  270°  ;  if  sec  x  =  80,  x  is  between  0°  and 
90°  or  between  270°  and  360°;  if  esc  x  =  -3,  x  is  between  180c 
and  360°. 

9.  In  Quadrant  III. ;  in  Quadrant  II.  ;  in  Quadrant  III. 

10.  40  angles  ;  20  positive  and  20  negative. 

11.  +,  when  x  is  known  to  be  in  Quadrant  I.  or  IV. ;  — ,  when  x  is  known 

to  be  in  Quadrant  II.  or  III. 

14.  sin  x  =  —  f  V3,     tan  x  =  -  4  V3,     cot  x  =»  -  ^2  V3,    esc  x  =  —  ^  V3. 

15.  sino;  =  ±TVVrTo,  coso;  =  =f  t3q  VIO,  tan  re—  -  \,  sec ar  =  =f  3  Vlo. 

esc  a;  =  ±  VlO. 

10.  The  cosine,  the  tangent,  the  cotangent,  and  the  secant  are  negative 
when  the  angle  is  obtuse. 

17.  Sine  and  cosecant  leave  it  doubtful  whether  the  angle  is  an  acute 
angle  or  an  obtuse  angle  ;  the  other  functions,  if  +  determine  an 
acute  angle,  if  —  an  obtuse  angle. 

20.  sin  450  -  sin  (360  +  90)  =  sin  90  =  1 ;  tan  540°  =  tan  180°  =  0  ; 
cos  630°  =  cos  270°  =  0  ;  cot  720°  =  cot  0°  =  oo ; 
sin  810°  -  sin  90°    =  1 ;  esc  900°  =  esc  180°  -  oo. 

21.  45°,  135°,  225°,  315°.  22.  0.  23.  0.  24.  0. 
25.   a2-Z>2  +  4a&. 

Exercise  X. 

2.  sin  172°-     sin    8°.  11.   cot  264°  =  tan  G°. 

3.  cos  100°  --  sin  10°.  12.   sec  244°  =  - esc  26°. 

4.  tan  125°  = -cot 35°.  13.   esc 271°  =  - sec  1°. 

5.  cot   91°  = -tan   1°.  14.   sin  163° 49'-     sin  16°  11'. 

6.  sec  110°  =  - esc  20°.  15.   cos  195°  33' =  - cos  15°  33'. 

7.  esc  157°=     esc  23°.  16.   tan  269°  15'=     cot  0°  45'. 

8.  sin  204° -- sin  24°.  17.   cot  139°  17'  =.  -cot 40°  43' . 

9.  cos  359°-     cos   1°.  18.   sec  299°  45'=      esc  29°  45'. 
10.  tan 300° --cot 30°.  19.  esc    92°  25' -      sec    2° 25'. 


ANSWERS. 


20.  sin(-  75°)= -sin   75°= -cos  15°,  cos(-  75°)=  cos  75°=    sin  15°,  etc. 

21.  sin(-127°)=  -sin  127°=  -cos  37°,  cos(-127°)=  cos  127°=  -sin  37°,  etc. 

22.  sin(-200°)=    sin  160°=    sin20o,cos(-200o)=cos200o=-cos20°,etc. 

23.  sin(-345°)  =  -sin  345°  =     sin  15°,  cos(-345°)  =  cos  345°  =  cos  15°.  etc. 

21.   sin(-    52°  37')  = -sin  52°  37'  = -cos 37°  23', 
cos  (-    52°  37')  -     cos  52°  37'  -     sin  37°  23',  etc. 

25.  sin  (-  196°  54')  =     sin  196°  54'=     cos  16°  54', 
cos  (-  196°  54')  =     cos  196°  54'  =  -  cos  16°  54',  etc. 

26.  sin  120°  =     £  V3,  cos  120°  =  -  h,  etc. 

27.  sin  135°  =  +  \  V2,  cos  1 35°  -  -  £  V2,  etc. 

28.  sin  150°  -  +  £,        cos  150°  =  -  £  V3,  etc. 

29.  sin  210°  -  -  J,        cos  210°  =  -  £  V3,  etc. 
B0.   sin  225°  -  - } V2,  cos 225°  =  -\y/%  etc. 

31.  sin  240°  =  -  J\/3,  cos  240°  -- 1,  etc. 

32.  sin  300°  --|V3,  cos 300°  -  +  \%  etc. 

33.  sin  (-30°)    --|,        cos (-30°)    =  +  %V3,  etc 

34.  sin  (-  225°)  =  +  h  V2,  cos  (-  225°)  -  - f V?  etc. 
35!  cos  x  =  -  I V2  or  -  VJ  etc.,  a  =  225°. 

36.  tan  x  =  —  \/|~   sin  x  —  J,  cos  #  =  —  \  y/S,  x  =  150°. 

37.  sin  3540°  -  sin  300°  =  -  sin  60°  -  -*|  V3,  cos  3540°  -  +  },  etc. 

38.  210°  and  330°  ;  120°  and  300°. 

89.   135°,  225°,  and  -225°  ;  150°  and  -30°. 

40.  30°,  150°,  390°,  and  510°. 

41.  sin  168°,   cos  334°,   tan  225°,   cot  252°, 
sin  349°,   cos  240°,   tan    64°,    cot  177°. 

42.  0.848.     (Hint:  tan 238°  =  tan 58°,  sin  122°  -  sin 58°). 

43.  -1.952.  47.  a2  +  62  +  2a6cosa;. 
44:  (a-b)smx.  48.   0. 

45.  m  sin  a?  cos  x.  49.   cos  x  sin  y  —  sin  x  cos  y. 

46.  (a  —  b)  cot  x  —  (a  +  b)  tan  x.  50.   tan  x. 

51.  Positive  between  x  =  0°  and  x  =  135°,  and  between  x  =  315°  and 

x  -  360°  ;  negative  between  x  =  135°  and  x  =  315°. 

52.  Positive  between  x  —  45°  and  x  =  225°  ;    negative  between  x  —  0° 

and  x  =  45°,  and  between  x  =  225°  and  x  =  360°. 

53.  sin  (x  —    90°)  =  —  cos  x,  cos  (a?  —    90°)  =  sin  x,  etc. 

54.  sin  (x  —  180°)  =  —  sin  x,  cos  (x  —  180°)  =  -  cos  x,  etc. 

Exercises  53  and  54  should  be  solved  by  drawing  suitable  figures,  and 
«mploying  a  mode  of  proof  sjmilar  to  that  used  in  \  28. 


TRIGONOMETRY. 


10.  S 

11.  S 

12.  s 

13.  s 

14.  S 

15.  s 

16.  s 

17.  i 


Exercise  XL 

n  (*  +  V)  =  ti  cos  (x  +  y)  =  ||. 
n(90°+y)  =     cosy,     cos  (  90  +  y)  =  —  sin  y,  etc. 
n(180—  y)=     sin y,     cos (180  —  y)  =  —cosy,  etc. 
n  (180  +  y)  =  —  sin  y,     cos  (180  +  y)  =  —  cos  y,  etc. 
n(270—  y)  =—  cosy,     cos  (270 -y)  =  -s:ny,  etc. 
n  (270  +  y)  =  -  cos  y,     cos  (270°  +  y)=     sin  y ,  etc. 
n(360°—  y)=—  siny,     cos (360  —  y)  =     cosy,  etc. 
n  (360  +  y)  =     sin  y,     cos  (360°  +  y)=     cos  y,  etc. 
n  (x  —  90°)  =  -  cos  x,    cos  (a;  —  90°)  =     sin  x,  etc. 
n  (x  -  180°)=  -  sin  z,    cos  (a  -  180°)=  -  cos  x,  etc. 
n  (x  -  270°)=     cos  #,    cos  (a;  —  270°)=  -sin  #,  etc. 
n(— 2/)         =— siny,     cos(— y)         =     cosy,  etc. 
n(45°-y)  =  £  V2 (cosy-sin y),     cos(45°-y)  =  J  V2(cosy+siny),etc. 
n(45°+y)  =  £  V2(cosy+siny),     cos (45°+y)  =  £  V2  (cosy-sin y), etc. 
n  (30°+y)  =  J  (cos  y  +  V%  sin  y),     cos  (30° +y)  =  \  ( V3  cos  y-sin  y),  etc. 
n  (60°— y)  =  \  ( V3  cos  y— sin  y),     cos  (60°— y)  =  \  (cos  y  +  V3  sin  y),  etc. 
18.   3  sin  a; -4  sin3  a;.         19.   4  cos3  a;- 3  cos  a;.         20.   0.         21.   £a/3. 

22.  sin  \x  ^J1  ~  0AV®  =  0.10051 ;     cos  £  a;  --J1  +  °'4  ^  -  0.99494. 

23.  cos2a;=--J,     tan2a?=»— V3. 

24.  sin  22£°  =  £  V2  -V2  =  0  3827,  cos  22£°  =  ^V2  +  V2  =  0.9239. 
tan  22 £°  -  V2  -1 =  0.4142,  cot  22£°  -  V2  +  1        =  2.4142. 

25.  sin  15°    =  £  V2  -  V3  =  0.2588,  cos  15°    -  ^V2+V3  -  0.9659. 
tan  15°    -  2  -  V3       -  0.2679,  cot  15°    =  2  +  V3       =  3.7321. 

27-33.  The  truth  of  these  equations  is  to  be  established  by  expressing 
the  given  functions  in  terms  of  the  same  function  of  the  same 
angle.     Thus,  in  Example  27, 

sin  2x  =  2sina;cosa;, 
and      2tanz  =  2 ,        1  +  tan2  x  =  sec2  x  =  — =r> 

COS  X  COS1*  X 

By  making  these  substitutions  in  the  given  equation  its  truth 
will  be  evident. 
34.   sin  A  +  sin  B  +  sin  C=  sin  A  +  sin  B  +  sin  [180  -  (A  +  B)] 
=  sin  A  +  sin  B  +  sin  (A  +  B) 

=  2sin£M-  +  ■#)  cos${A  -  B)  +  2sinJ(-4  +  B)  co$$(A  +  B) 
«  2sin*U  +  #)  [cos  J (4  -  B)  +  cos$(A  +  B)] 
=  4  sin  \  (A  +  B)  cos  £  A  cos  \B,  (see  g  \  34  and  35) 

But  cosJC=  cos [90°-  \{A  +  B)]  =  Bm%(A  +  B). 

Therefore,  sin  A  +  sin  B  +  sin  C=~  4  cos  \  A  cos  |  B  cos  \  C. 


ANSWERS. 


35.  Proof  similar  to  that  for  34. 

00    .       a  ,  .      t>  ,  l      n  smAcosBcosA  sin  B     sin  0 

36.  tan  A  +  tan B  +  tan  C  =  — — .  + _  + _ 

cos  A  cos  B     cos  A  cos  B     cos  0 

sin  C  sin  0  sin  C  cos  C+  cos  A  cos  i?  sin  C 


+ 


cos  A  cos  i?     cos  CY  cos  A  cos  B  cos  C 

(cos  AcosB  +  cos  C)  sin  C_  [cos  A  cos  5  —  cos  (A  +  i?)]  sin  C 
cos  J.  cos  i?  cos  0  cos  J.  cos  B  cos  C 


=  tan  A  tan  5  tan  C 


_  sin  A  sin  5  sin  C 
cos  A.  cos  B  cos  C 
37.   Proof  similar  to  that  for  36. 
oo         2  42.    tan2  x.  .„     cos  (#  +  y) 

sin2a:  .„     cos(#— y)  '    sinrcsiny 

39.    2 cot  2 x.  cos  a;  cosy  47.   tan  x  tan  y. 

40#    cos(a:-y)  u     cosp-f  y) 

sin  a;  cos  y  cos  #  cos  y 

41.   cos(x  +  y)_  45.    cos(a;-y) 

sin  a?  cosy  '    sin  x  sin  y 

Exercise  XII. 

1.   If,  for  instance,  B=  90°,  [25]  becomes  -  -  sin  A. 

b 
3.   a2  =  62  +  c2,     a2  =  62  +  c2-26c,     a2  =  62  +  c2  +  26c. 

6.  90°  in  each  case. 

7.  (i.)    ^^  -  tan  (J.  -  45°),  and  a  right  triangle. 

(ii.)    a  +  6  =  (a  —  6)  (2  +  V3),  an  isosceles  triangle  with  the  angles  30c 
30°,  120°. 


9.    300. 

10.  AB  =  59.564  miles  ; 
AC  =  54.285  miles. 

11.  4.6064  miles,  4.4494  m 
3.7733  miles. 

12.  4.1501  and  8.67. 

13.  6.1433  miles  and 

14.  8  and  5.4723. 


Exercise  XIV. 
11.   420.         12.   The  other  diagonal  -  124.617 


EXERC] 

[SE 

XIII. 

15. 

a  =  5,  c  =  9.6592. 

16. 

a  =  7,  6=8.573. 

lies, 

17. 

Sides,  600  feet  and  1039.2  feet 
altitude,  519.6  feet. 

18. 

855 :  1607. 

8  miles. 

19. 

20. 

5.438  and  6.857. 
15.588. 

10 


TRIGONOMETRY. 


11.  6. 

12.  10.392. 
14.   8.9212. 


Exercise  XV. 


15.  25. 

16.  3800  yards. 

17.  729.7  yards. 


18.  10.266. 

19.  a  =  5.0032,  b  =  2.3385. 

20.  26°0'10"ar1dl4o5'50' 


11.  4 -36°  5*  12",  B 

12.  ^4  =  £  =  33°33'27 

13.  A  =  B=C=60°. 

14.  Impossible. 

15.  15°,  45°,  120°. 


Exercise  XVI. 

=  53°  7'  4S",  C  =  90°.        16.  45°,  60°,  75°. 
G'=  112° 53' 0".    17.   4°  23'  W.  of  N.,  or  W.  of  S. 
18.   60°. 

20.  0.88873. 

21.  54.516  miles. 


Exercise  XVII, 


1.  4333600. 

2.  365.68. 

3.  13260. 

4.  8160. 

5.  240. 

6.  26208. 

7.  15540. 

8.  29450  or  6983. 


9.  10  V3  =  17.3205. 

10.  6V3  =  10.3923. 

11.  0.19952. 

12.  ab  sin  A. 

13.  £(a2-Z>2)tanA 

14.  2421000. 

15.  30°.  30°.  120°. 


1.  21.166  miles 

2.  6.3399  miles. 

3.  119.29  feet. 

4.  30°. 


Exercise  XVITT 
24.966  miles. 


5.  20  feet. 

6.  2.6268  or  21.4704. 

7.  276.14  yards. 

8.  383.35  yards. 


ANSWERS.  11 


Miscellaneous  Problems. 


2. 

107  feet ;  143  feet.  27. 

8  inches. 

51. 

757.5  feet. 

3. 

1024  feet. 

30. 

460.45  feet. 

52. 

520  yards. 

4. 

37°  44'  5". 

31. 

88.94  feet. 

53. 

1366.4  feet. 

5. 

238,400  miles. 

32. 

13.657  miles. 

54. 

658  pounds ; 

6 

861,800  miles. 

34. 

56.5  feet. 

22°  24'  with  first 

7. 

2922.4  miles. 

35. 

51.6  feet 

force. 

8. 

60°. 

36. 

101.89  feet. 

55. 

88.33  pounds ; 

9. 

3.2. 

38. 

N.  76°56'E.; 

45°  37'  with  known 

10. 

6.6. 

13.94  miles  an 

hour. 

force. 

11. 

199.56  feet. 

39. 

422  yards. 

58. 

500.2;  536.3. 

12. 

43  feet. 

40. 

256  feet. 

59. 

345.47  feet. 

13. 

45  feet. 

41. 

3121  feet ; 

60. 

345.47  yards. 

14. 

26°  34'. 

3633.5  feet. 

61. 

61.23  feet. 

15. 

78.37  feet. 

42. 

529.5  feet. 

63. 

307.79. 

16. 

75  feet. 

43. 

41.41  feet. 

64. 

19.8;  35.7;  44.5. 

17. 

1.44  miles. 

44. 

234.5  feet. 

65. 

45°. 

19. 

56.65  feet. 

45. 

25.43  miles. 

68. 

60°. 

20. 

69.28  feet. 

46. 

294.7  feet. 

69. 

60°. 

21. 

260  feet ;  3690  feet.  47. 

12,492  feet. 

70. 

30°. 

22. 

1.344  miles. 

48. 

6.34  miles. 

73. 

$  be  sin  A. 

23. 

235.8  yards. 

49. 

210.44  feet. 

74. 

\  c2  sin  A  sin  B  esc  (.4 

+  B). 

75. 

VK< 

5  —  O)  (S  —  b)  (S  — 

c)l 

77. 

199  a.  3  b.  10  p. 

94. 

16,281. 

114. 

S.  56°  7' 30"  E. ; 

78. 

210  a.  3  e.  26  p. 

95. 

435.8  feet. 

202.6  miles. 

79. 

12  a.  3  e.  37  p. 

96. 

49,089  feet. 

115. 

N.  17°25'W.; 

80. 

3  a.  0  e.  6  p. 

97. 

750  feet. 

37°  46'  N. 

81. 

12  A.  1  E.  14  P. 

98. 

422.4  feet. 

116. 

56°  11'  E. ;  244.3 

82. 

4  a.  2  e.  26  p. 

99. 

1835  feet. 

121. 

Long.  68°  55'  W. 

83. 

14  a.  2  e.  9  p. 

100. 

26.88. 

122. 

103.6  miles. 

84. 

61  A.  2  E. 

103. 

6. 

124. 

33°  18'  N.  ; 

85. 

4  A.  2  e.  26  p. 

108. 

6. 

36°  24'  W. 

86. 

13.93,  23.21.32.5ch.no. 

6087  feet. 

125. 

N.  28°47'K; 

87. 

9  a. 

111. 

5°25'S.; 

1293  miles. 

89. 

876.31. 

457.5  miles. 

126. 

S.  50°  40'  W. ; 

90. 

1229.5. 

112. 

460.8;  383.1  miles. 

250.8 ;  20°  9'  W. 

92. 

1075.3. 

113. 

229  miles ; 

127. 

38°  21'  N. ; 

93. 

2660.45. 

lat.  11°  39'  S. 

55°12'W. 

12  TRIGONOMETRY. 


128.   171  miles  ;  32°  44'  W.  129.  N.  36°  52'  W. ;  36°  8'  W. 

130.  173  miles ;  51°  16'  S. ;  34°  13'  E. 

131.  S.  50°  58'  E. ;  47°  W  N.;  20°  49'  W. 

132.  N.  53°  20'  E.,  16°  7'  W.;  or  N.  53°  2W  W.,  25°  53'  W 
133.  N.  47°  42^  E.,  19°  27'  N.,  121°  51'  E. ;  or  N.  47°  42.5'  W.,  19°  27* 

N.,  116°  9'  E. ;  or  S.  47°  42.5'  E.,  14°  33'  N.,  121°  48'  E. ;  or  S 
47°  42.5'  W.,  14°  33'  N.,  116°  12'  E. 

137.  N.  73'  E.,  45  miles ;  42°  15*  N.,  69°  &  W. 

138.  N.  72°  W.,  287  miles :  33°  S..  13°  2'  E 


FIVE-PLACE 


LOGARITHMIC   AND   TRIGONOMETRIC 

TABLES. 


ARRANGED    BY 

G.    A    WENTWORTH,    A.M., 

AND 

G.    A    HILL,    A.M. 


>'**c 


BOSTON,  U.S.A.: 

PUBLISHED    BY   GINN   &   COMPANY. 

1891. 


Entered  according  to  Act  of  Congress,  in  the  year  1882,  by 

G.  A.  WENTWORTH  and  G.  A.  HILL, 
in  the  office  of  the  Librarian  of  Congress  at  Washington. 


J.  S.  Cushing  &  Co.,  Printers,  Boston. 


INTRODUCTION. 


>^< 


1.  If  the  natural  numbers  are  regarded  as  powers  of  ten,  the  ex- 
ponents of  the  powers  are  the  Common  or  Briggs  Logarithms  of  the 
numbers.  If  A  and  B  denote  natural  numbers,  a  and  b  their  loga- 
rithms, then  10a  =  A,  105  =  B  ;  or,  written  in  logarithmic  form, 

log^l  =  «,         \ogB=b. 


2.  The  logarithm  of  a  product  is  found  by  adding  the  logarithms 
of  its  factors. 

For,  AxB   =  10«  X  106  =  10*  +  \ 

Therefore,  log  ( A  x  B)  =  a  +  b  =  log  A  +  log  B. 

3.  The  logarithm  of  a  quotient  is  found  by  subtracting  the  logarithm 
of  the  divisor  from  that  of  the  dividend. 

For.  4.«g,.M~>. 

B>    10* 

A 

Therefore,  log  —  =  a  —  b  =  log  A  —  log  B. 

4.  The  logarithm  of  a  power  of  a  number  is  found  by  multiplying 
the  logarithm  of  the  number  b}^  the  exponent  of  the  power. 

For,  An=(10a)n=10an. 

Therefore,  log  An  =  an  —  n  log  A. 


5.   The  logarithm  of  the  root  of  a  number  is  found  by  dividing  the 
logarithm  of  the  number  b}r  the  index  of  the  root. 


For,  VJ  =  \/l0«  =  10* 

logA 
n 


Therefore,  log  \/~A.  =  -  = 


6.  The  logarithms  of  1,  10,  100,  etc.,  and  of  0.1,  0.01,  0.001,  etc, 
are  integral  numbers.  The  logarithms  of  all  other  numbers  are  frac- 
tions. 


IV  LOGARITHMS. 


For,    10°=       1,  hence       log  1  =  0;  10-!=     0.1,  hence     log  0.1  =  —1; 

101  =     10,  hence     log  10  =  1 ;  10~2  =   0.01,  hence    log  0.01  =  —2 ; 

102=    100,  hence    logl00  =  2;  10"3  =  0.001,  hence  log  0.001  = -3; 

103  =  1000,  hence  log  1000  =  3 ;  and  so  on. 

If  the  number  is  between  1  and  10,  the  logarithm  is  between  0  and  1. 
If  the  number  is  between  10  and  100,  the  logarithm  is  between  1  and  2. 
If  the  number  is  between  100  and  1000,  the  logarithm  is  between  2  and  3. 
If  the  number  is  between  1  and  0.1,  the  logarithm  is  between  0  and  —1. 
If  the  number  is  between  0.1  and  0.01,  the  logarithm  is  between  —1  and  —2. 
If  the  number  is  between  0.01  and  0.001,  the  logarithm  is  between  —2  and  —3. 
And  so  on. 

7.  If  the  number  is  less  than  1,  the  logarithm  is  negative  (§6), 
but  is  written  in  such  a  form  that  the  fractional  part  is  always  positive. 

For  the  number  may  be  regarded  as  the  product  of  two  factors,  one  of 
which  lies  between  1  and  10,  and  the  other  is  a  negative  power  of  10 ;  the 
logarithm  will  then  take  the  form  of  a  difference  whose  minuend  is  a  positive 
proper  fraction,  and  whose  subtrahend  is  a  positive  integral  number. 

Thus,  0.48  =  4.8X0.1. 

Therefore  (§2),  log     0.48  =  log 4.8  +  logO.l  =  0.68124-1.     (Page  1.) 

Again,  0.0007  =  7X0.0001. 

Therefore,  log  0.0007  =  log  7  +  log 0.0001  =  0.84510  -  4. 

8.  Every  logarithm,  therefore,  consists  of  two  parts :  a  positive 
or  negative  integral  number,  which  is  called  the  Characteristic,  and 
a  positive  proper  fraction,  which  is  called  the  Mantissa. 

Thus,  in  the  logarithm  3.52179,  the  integral  number  3  is  the  characteristic, 
and  the  fraction  .52179  the  mantissa.  In  the  logarithm  0.78254  —  2,  the  inte- 
gral number  —  2  is  the  characteristic,  and  the  fraction  .78254  is  the  mantissa. 

9.  If  the  logarithm  is  negative,  it  is  customary  to  change  the  form 
of  the  difference  so  that  the  subtrahend  shall  be  10  or  a  multiple  of  10. 
This  is  done  by  adding  to  both  minuend  and  subtrahend  a  number 
which  will  increase  the  subtrahend  to  10  or  a  multiple  of  10. 

Thus,  the  logarithm  0.78254-2  is  changed  to  8.78254-10  by  adding  8  to 
both  minuend  and  subtrahend.  The  logarithm  0.92737  —  13  is  changed  to 
7.92737  —  20  by  adding  7  to  both  minuend  and  subtrahend. 

10.  The  following  rules  are  derived  from  §  6  :  — 

If  the  number  is  greater  than  1,  make  the  characteristic  of  the 
logarithm  one  unit  less  than  the  number  of  figures  on  the  left  of  the 
decimal  point. 

If  the  number  is  less  than  1 ,  make  the  characteristic  of  the  loga- 
rithm negative,  and  one  unit  more  than  the  number  of  zeros  between 
the  decimal  point  and  the  first  significant  figure  of  the  given  number. 


INTRODUCTION. 


If  the  characteristic  of  a  given  logarithm  is  positive,  make  the 
number  of  figures  in  the  integral  part  of  the  corresponding  number 
one  more  than  the  number  of  units  in  the  characteristic. 

If  the  characteristic  is  negative,  make  the  number  of  zeros  between 
the  decimal  point  and  the  first  significant  figure  of  the  corresponding- 
number  one  less  than  the  number  of  units  in  the  characteristic. 

Thus,  the  characteristic  of  log  7849.27  =  3 ; 

the  characteristic  of  log  0.037  =  —  2  =  8.00000  —  10. 
If  the  characteristic  is  4,  the  corresponding  number  has  five  figures  in  its 
integral  part.     If  the  characteristic  is  —  3,  that  is,  7.00000  —  10,  the  corre- 
sponding fraction  has  two  zeros  between  the  decimal  point  and  the  first 
significant  figure. 

11.  The  logarithms  of  numbers  that  can  be  derived  from  one 
another  b}r  multiplication  or  division  by  an  integral  power  of  10  have 
the  same  mantissa. 

For,  multiplying  or  dividing  a  number  by  an  integral  power  of  10  will 
increase  or  diminish  its  logarithm  by  the  exponent  of  that  power  of  10 ;  and 
since  this  exponent  is  an  integer,  the  mantissa  of  the  logarithm  will  be 
unaffected. 

Thus,        log  4.6021     =0.66296.     (Page  9.) 

log  460.21     =  log  (4.6021  X  102)  =  log  4.6021  +  log  102 

=  0.66296  +  2  =  2.66296. 
log  460210    =  log  (4.6021  X  105)  =  log  4.6021  +  log  105 

=  0.66296  +  5  =  5.66296. 
log  0.046021  =  log  (4.6021  -*■  102)  =  log  4.6021  -  log  10' 

=  0.66296  -  2  =  8.66296  -  10. 


TABLE    I. 

12.  In  this  table  (pp.  1-19)  the  vertical  columns  headed  N  contain 
the  numbers,  and  the  other  columns  the  logarithms.  On  page  1  both 
the  characteristic  and  the  mantissa  are  printed.  On  pages  2-19  the 
mantissa  only  is  printed. 

The  fractional  part  of  a  logarithm  can  be  expressed  only  approxi- 
mately, and  in  a  five-place  table  all  figures  that  follow  the  fifth  are 
rejected.  Whenever  the  sixth  figure  is  5,  or  more,  the  fifth  figure  is 
increased  by  1.  The  figure  5  is  written  when  the  value  of  the  figure 
in  the  place  in  which  it  stands,  together  with  the  succeeding  figures,  is 
more  than  4£,  but  less  than  5. 

Thus,  if  the  mantissa  of  a  logarithm  written  to  seven  places  is  5328732, 
it  is  written  in  this  table  (a  five-place  table)  53287.  If  it  is  5328751,  it  is 
written  53288.     If  it  is  5328461  or  5328499,  it  is  written  in  this  table  53285. 

Again,  if  the  mantissa  is  5324981,  it  is  written  53250;  and  if  it  is  4999967, 
it  is  written  50000. 


VI  LOGARITHMS. 


This  distinction  between  5  and  5,  in  case  it  is  desired  to  curtail  still 
further  the  mantissas  of  logarithms,  removes  all  doubt  whether  a  5  in 
the  last  given  place,  or  in  the  last  but  one  followed  b}-  a  zero,  should 
be  simply  rejected,  or  whether  the  rejection  should  lead  us  to  increase 
the  preceding  figure  by  one  unit. 

Thus,  the  mantissa  13925,  when  reduced  to  four  places  should  be  1392; 
but  13925  should  be  1393. 

To  Find  the  Logarithm  of  a  Given  Number. 

13.  If  the  given  number  consists  of  one  or  two  significant  figures, 
the  logarithm  is  given  on  page  1.  If  zeros  follow  the  significant 
figures,  or  if  the  number  is  a  proper  decimal  fraction,  the  character- 
istic must  be  determined  b}r  §  10. 

14.  If  the  given  number  has  three  significant  figures,  it  will  be 
found  in  the  column  headed  N  (pp.  2-19),  and  the  mantissa  of  its 
logarithm  in  the  next  column  to  the  right,  and  on  the  same  line. 
Thus, 

Page    2.    log  Ho  =  2.16137,  log  14500  =  4.16137. 

Page  14.    log716  =  2.85491,  log  0.716  =  9.85491  -  10. 

15.  If  the  given  number  has  four  significant  figures,  the  first  three 
will  be  found  in  the  column  headed  N,  and  the  fourth  at  the  top  of 
the  page  in  the  line  containing  the  figures  1,  2,  3,  etc.  The  mantissa 
will  be  found  in  the  column  headed  by  the  fourth  figure,  and  on  the 
same  line  with  the  first  three  figures.     Thus, 

Page  15.    log  7682   =3.88547,        log  76.85    =1.88564. 
Page  18.    log  93280  =  4.96979,         log  0.9468  =  9.97626  -  10. 

16.  If  the  given  number  has  five  or  more  significant  figures,  a 
process  called  interpolation  is  required. 

Interpolation  is  based  on  the  assumption  that  between  two  con- 
secutive mantissas  of  the  table  the  change  in  the  mantissa  is  directly 
proportional  to  the  change  in  the  number. 

Required  the  logarithm  of  34237. 

The  required  mantissa  is  (§  11)  the  same  as  the  mantissa  for  3423.7 ;  there- 
fore it  will  be  found  by  adding  to  the  mantissa  of  3423  seven-tenths  of  the 
difference  between  the  mantissas  for  3423  and  3424. 

The  mantissa  for  3423  is  53441. 

The  difference  between  the  mantissas  for  3423  and  3424  is  12. 

Hence,  the  mantissa  for  3423.7  is  53441  +  (0.7  X  12)  =  53449. 

Therefore,  the  required  logarithm  of  34237  is  4.53449. 


INTRODUCTION.  Vll 


Required  the  logarithm  of  0.0015764. 

The  required  mantissa  is  the  same  as  the  mantissa  for  1576.4;  therefore 
it  will  be  found  by  adding  to  the  mantissa  for  1576  four-tenths  of  the  difference 
between  the  mantissas  for  1576  and  1577. 

The  mantissa  for  1576  is  19756. 

The  difference  between  the  mantissas  for  1576  and  1577  is  27. 

Hence,  the  mantissa  for  1576.4  is  19756  +  (0.4  x  27)  =  19767. 

Therefore,  the  required  logarithm  of  0.0015764  is  7.19767-10. 

Required  the  logarithm  of  32.6708. 

The  required  mantissa  is  the  same  as  the  mantissa  for  3267.08 ;  therefore 
it  will  be  found  by  adding  to  the  mantissa  for  3267  eight-huudredths  of  the 
difference  between  the  mantissas  for  3267  and  3268. 

The  mantissa  for  3267  is  51415. 

The  difference  between  the  mantissas  for  3267  and  3268  is  13. 

Hence,  the  mantissa  for  3267.08  is  51415  +  (0.08  X  13)  =  51416. 

Therefore,  the  required  logarithm  of  32.6708  is  1.51416. 


17.  When  the  fraction  of  a  unit  in  the  part  to  be  added  to  the 
mantissa  for  four  figures  is  less  than  0.5  it  is* to  be  neglected;  when 
it  is  0.5  or  more  than  0.5  it  is  to  be  taken  as  one  unit. 

Thus,  in  the  first  example,  the  part  to  be  added  to  the  mantissa  for  3423 
Is  8.4,  and  the  .4  is  rejected.  In  the  second  example,  the  part  to  be  added  to 
the  mantissa  for  1576  is  10.8,  and  11  is  added. 


To  Find  the  Number  Corresponding  to  a  Given  Logarithm. 

18.  If  the  given  mantissa  can  be  found  in  the  table,  the  first  three 
figures  of  the  required  number  will  be  found  in  the  same  line  with  the 
mantissa  in  the  column  headed  N,  and  the  fourth  figure  at  the  top  of 
the  column  containing  the  mantissa. 

The  position  of  the  decimal  point  is  determined  by  the  character- 
istic (§10). 

Find  the  number  corresponding  to  the  logarithm  0.92002. 

Page  16.     The  number  for  the  mantissa  92002  is  8318. 
Therefore,  the  required  number  is  8.318. 

Find  the  number  corresponding  to  the  logarithm  6.09167. 

Page  2.     The  number  for  the  mantissa  09167  is  1235. 
Therefore,  the  required  number  is  1235000. 

Find  the  number  corresponding  to  the  logarithm  7.50325  —  10. 
Page  6.    The  number  for  the  mantissa  50325  is  3186. 
Therefore,  the  required  number  is  0.003186. 


Vlll  LOGARITHMS. 


19.  If  the  given  mantissa  cannot  be  found  in  the  table,  find  in  the 
table  the  two  adjacent  mantissas  between  which  the  given  mantissa 
lies,  and  the  four  figures  corresponding  to  the  smaller  of  these  two 
mantissas  will  be  the  first  four  significant  figures  of  the  required 
number.  If  more  than  four  figures  are  desired,  they  may  be  found  by 
interpolation,  as  in  the  following  examples  : 

Find  the  number  corresponding  to  the  logarithm  1.48762. 

Here  the  two  adjacent  mantissas  of  the  table,  between  which  the  given 
mantissa  48762  lies,  are  found  to  be  (page  6)  48756  and  48770.  The  corre- 
sponding numbers  are  3073  and  3074.  The  smaller  of  these,  3073,  contains 
the  first  four  significant  figures  of  the  required  uumber. 

The  difference  between  the  two  adjacent  mantissas  is  14,  and  the  difference 
between  the  corresponding  numbers  is  1. 

The  difference  between  the  smaller  of  the  two  adjacent  mantissas,  48756, 
and  the  given  mantissa,  48762,  is  6.  Therefore,  the  number  to  be  annexed  to 
3073  is  T6T  of  1  =  0.428,  and  the  fifth  significant  figure  of  the  required  number 
is  4. 

Hence,  the  required  number  is  30.734. 

Find  the  number  corresponding  to  the  logarithm  7.82326  —  10. 

The  two  adjacent  mantissas  between  which  82326  lies  are  (page  13)  82321 
and  82328.     The  number  corresponding  to  the  mantissa  82321  is  6656. 

The  difference  between  the  two  adjacent  mantissas  is  7,  and  the  difference 
between  the  corresponding  numbers  is  1. 

The  difference  between  the  smaller  mantissa,  82321,  and  the  given  mantissa, 
82326,  is  5.  Therefore,  the  number  to  be  annexed  to  6656  is  \  of  1  =  0.7,  and 
the  fifth  significant  figure  of  the  required  number  is  7. 

Hence,  the  required  number  is  0.0066567. 

In  using  a  five-place  table  the  numbers  corresponding  to  mantissas 
may  be  carried  to  five  significant  figures,  and  in  the  first  part  of  the 
table  to  six  figures.* 

20.  The  logarithm  of  the  reciprocal  of  a  number  is  called  the 
Cologarithm  of  the  number. 

If  A  denote  any  number,  then 

cologJ.  =  log^-  =  logl  —  log  A  (§  3)  =  — log^l. 
A 

Hence,  the  cologarithm  of  a  number  is  equal  to  the  logarithm  of 
the  number  with  the  minus  sign  prefixed,  which  sign  affects  the  entire 
logarithm,  both  characteristic  and  mantissa. 

*  In  most  tables  of  logarithms  proportional  parts  are  given  as  an  aid  to 
Interpolation;  but,  after  a  little  practice,  the  operation  can  be  performed 
nearly  as  rapidly  without  them.  Their  omission  allows  a  page  with  larger- 
faced  type  and  more  open  spacing,  and  consequently  less  trying  to  the  eyes. 


INTRODUCTION.  IX 


In  order  to  avoid  a  negative  mantissa  in  the  cologarithm,  it  is 
customary  to  substitute  for  —  log  A  its  equivalent 

(lO-log^)-lO. 

Hence,  the  cologarithm  of  a  number  is  found  by  subtracting  the 
logarithm  of  the  number  from  10,  and  then  annexing  —  10  to  the 
remainder. 

The  best  way  to  perform  the  subtraction  is  to  begin  on  the  left  and 
subtract  each  figure  of  log  ^4.  from  9  until  we  reach  the  last  significant 
figure,  which  must  be  subtracted  from  10. 

If  log^l  is  greater  in  absolute  value  than  10  and  less  than  20,  then 
in  order  to  avoid  a  negative  mantissa,  it  is  necessary  to  write  —  log  A 
in  the  form 

(20-log-4)-20. 

So  that,  in  this  case,  colog^.  is  found  by  subtracting  log^.  from  20, 
and  then  annexing  —  20  to  the  remainder. 

Find  the  cologarithm  of  4007. 

10  —10 

Page  8.  log  4007=   3.60282 

colog4007  =    6.39718-10 

Find  the  cologarithm  of  103992000000. 

20  -20 

Page  2.     log  103992000000  =  11 .01700 

colog  103992000000  =    8.98300  -  20 

If  the  characteristic  of  log  A  is  negative,  then  the  subtrahend,  —10 
or  —  20,  will  vanish  in  finding  the  value  of  colog-<4. 


Find  the  cologarithm  of  0.004007. 

10  -10 

log  0.004007  =    7.60282  -  10 


colog  0.004007  =    2.39718 

With  practice,  the  cologarithm  of  a  number  can  be  taken  from  the 
table  as  rapidly  as  the  logarithm  itself. 

By  using  cologarithms  the  inconvenience  of  subtracting  the  loga- 
rithm of  a  divisor  is  avoided.  For  dividing  by  a  number  is 
equivalent  to  multiplying  by  its  reciprocal.  Hence,  instead  of 
subtracting  the  logarithm  of  a  divisor  its  cologarithm  may  be  added. 


X 

LOGAR1THMS. 

Computation  by  Logarithms. 

21.(1) 

Find  the  value  of  x,  if  x  =  72214  x  0.08203. 

Page  14.           log  72214      =  4.85862 
Page  16.           log  0.08203   =8.91397-10 

By  §2.              logx               =3.77259 
Page  11.                 x              =  5923.63 

(2) 

Find  the  value  of  x,  if  x  =  5250  -j-  23487. 

Page  10.           log  5250         =3.72016 
Page  4.         colog  23487       =  5.62917  -  10 

Page  4.             log x              =  9.34933  —  10  =  log  0.22353 
.-.    x              =0.22353 

(3) 

Find  the  value  of  x   if  x  -        7'56  X  4667  X  567 

Find  the  value  of  z,  if  x-  ^  ^  ^^  ^  ^ 

Page  15.           log  7.56         =  0.87852 
Page  9.        .     log  4667         =3.66904 
Page  11.           log  567          =2.75358 
Page  17. '     colog  899. 1       =  7.04619  -  10 
Page  6.         colog  0.00337    =  2.47237 
Page  4.         colog  23435       =  5.63013  -  10 

Page  5.             log  x              =  2.44983  =  log  281.73 
.-.   x              =281.73 

(4) 

Find  the  cube  of  376. 

Page  7.             log  376           =2.57519 
Multiply  by  3  (§4),                           3 

Page  10.           log  3763          =  7.72557  =  log  53158600 
.-.    3763          =53158600 

(5) 

Find  the  square  of  0.003278. 

Page  6.             log  0.003278  =    7.51561  -  10 

2 

Page  2.             log  0.0032782=  15.03122  -  20  =  log  0.000010745 
.-.    0.0032782=  0.000010745 

(6) 

Find  the  square  root  of  8322. 

Page  16.            log  8322          =  3.92023 
Divide  by  2  (§  5),                 2)  3.92023 

logV8322      =1.96012  =  log  91.226 

.-.    V8322      =91.226 

If  the  given  number  is  a  proper  fraction,  its  logarithm  will  have  as 
a  subtrahend  10  or  a  multiple  of  10.     In  this  case,  before  dividing  the 
logarithm  by  the  index  of  the  root,  both  the  subtrahend  and  the  num- 

INTRODUCTION.  -  XI 


ber  preceding  the  mantissa  should  be  increased  b}T  such  a  number  as 
will  make  the  subtrahend,  when  divided  by  the  index  of  the  root, 
10  or  a  multiple  of  10. 

(7)  Find  the  square  root  of  0.000043641. 

Page  8.     log  0.000043641     =    5.63989-10 

10  -10 

Divide  by  2  (§  5>, 2)15.63989-^20 

Page  13.    log  vO, 000043641  =    7.81995  -  10  =  log  0.0066062 
.-.     V0.000043641  =    0.0066062 

(8)  Find  the  sixth  root  of  0.076553. 

Page  15.    log  0.076553  =    8.88397-10 

50  -50 


Divide  by  6  (§  5), 6)58.88397-60 

Page  13    log  a/0.076553       =    9.81400  -  10  =  log  0.65163 
.-.     ^0.076553       =    0.65163 


TABLE    II. 

22.   This  table  (page  20)  contains  the  value  of  the  number  tt,  its 
most  useful  combinations,  and  their  logarithms. 

Find  the  length  of  an  arc  of  47°  32'  57"  in  a  unit  circle. 

47°  32' 57"  =171177" 
log  171177  =5.23344 

log-i-  =4,68557-10 


log  arc  47°  32'  57"  =  9.91901  -  10  =  log  0.82994 
.  \  length  of  arc      =  0.82994 

Find  the  angle  if  the  length  of  its  arc  in  a  unit  circle  =  0.54936. 
log  0.54936  =  9.73986  -  10 

colog-L^loga"  =5.31443 
a"  


log  angle  =  5.05429  =  log  113316 

.-.    angle  =  113316"=  31°  28' 36" 

23.   The  relations  between  arcs  and  angles  given  in  Table  II.  are 
readily  deduced  from  the  circular  measure  of  an  angle. 

In  Circular  Measure  an  angle  is  defined  by  the  equation 

angle  =  ^5_, 
radius 

in  which  the  word  arc  denotes  the  length  of  the  arc  corresponding  to 
the  angle,  when  both  arc  and  radius  are  expressed  in  terms  of  the 
same  linear  unit. 


Xll  LOGARITHMS. 


Since  the  arc  and  radius  for  a  given  angle  in  different  circles  vary 
in  the  same  ratio,  the  value  of  the  angle  given  by  this  equation  is 
independent  of  the  value  of  the  radius. 

The  angle  which  is  measured  by  a  radius-arc  is  called  a  Radian, 
and  is  the  angular  unit  iu  circular  measure. 

Since  C=  2ttR,  it  follows  that-  =  2tt,  and  i^  =  tt.     Therefore, 

R  R 

If  the  arc  =  circumference,  the  angle  =  Zir. 

If  the  arc  =  semicircumference,  the  angle  =  w. 
If  the  arc  =  quadrant,  the  angle  =  %ir. 

If  the  arc  =  radius,  the  angle  =  1. 

Therefore,  tt  =  180°,    ^=90°,    -^=60°,    i*  =  45°,    1^  =  30°, 
\ir  =  221°,  and  so  on. 

Since  180°  in  common  measure  equals  ir  units  in  circular  measure, 
1°  in  common  measure      =  -~-  units  in  circular  measure  ; 

,       u  •      •      i  180°  . 

1  unit  in  circular  measure  = in  common  measure. 

7T 

By  means  of  these  two  equations,  the  value  of  an  angle  expressed 
in  one  measure  may  be  changed  to  its  value  in  the  other  measure. 

Thus,  the  angle  whose  arc  is  equal  to  the  radius  is  an  angle  of 

1  S0° 
1  unit  in  circular  measure,  and  is  equal  to  ,  or  57°  17'  45",  very 

7T 

nearly. 

TABLE    III. 

24.  This  table  (pp.  21-49)  contains  the  logarithms  of  the  trigo- 
nometric functions  of  angles.  In  order  to  avoid  negative  character- 
istics, the  characteristic  of  every  logarithm  is  printed  10  too  large. 
Therefore,  —  10  is  to  be  annexed  to  each  logarithm. 

On  pages  28-49  the  characteristic  remains  the  same  throughout 
each  column,  and  is  printed  at  the  top  and  the  bottom  of  the  column. 
But  on  pp.  30,49,  the  characteristic  changes  one  unit  in  value  at  the 
places  marked  with  bars.  Above  these  bars  the  proper  characteristic 
is  printed  at  the  top,  and  below  them  at  the  bottom,  of  the  column. 

25.  On  pages  28-49  the  log  sin,  log  tan,  log  cot,  and  log  cos,  of  1° 
to  89°,  are  given  to  every  minute.  Conversely,  this  part  of  the  table 
gives  the  value  of  the  angle  to  the  nearest  minute  when  log  sin,  log  tan, 
log  cot,  or  log  cos  is  known,  provided  log  sin  or  log  cos  lies  between 
8.23822  and  9.99992,  and  log  tan  or  log  cot  lies  between  8.23829  and 
11.76171. 


INTRODUCTION.  Xlll 


If  the  exact  value  of  the  given  logarithm  of  a  function  is  not  found 
in  the  table,  the  value  nearest  to  it  is  to  be  taken,  unless  interpolation 
is  employed  as  explained  in  §  26. 

If  the  angle  is  less  than  45°,  the  number  of  degrees  is  printed  at 
the  top  of  the  page,  and  the  number  of  minutes  in  the  column  to  the 
left  of  the  columns  containing  the  logarithm.  If  the  angle  is  greater 
than  45°,  the  number  of  degrees  is  printed  at  the  bottom  of  the  page, 
and  the  number  of  minutes  in  the  column  to  the  right  of  the  columns 
containing  the  logarithms. 

If  the  angle  is  less  than  45°,  the  names  of  its  functions  are  printed 
at  the  top  of  the  page ;  if  greater  than  45°,  at  the  bottom  of  the 
page.     Thus, 

Page  38.   log  sin  21°  37'  =    0.56631-10. 

Page  45.    log  cot  36°  53'  =  10. 12473  —  10  =  0. 12473. 

Page  37.  log  cos  69°  14' =    9.54969-10. 

Page  49.    log  tan  45°  59'  =  10.01491  -  10  =  0.01491. 

Page  48.    If  log  cos  =  9.87468  —  10,  angle  =  41°  28'. 

Page  34.    If  log  cot  =  9.39353  —  10,  angle  =  76°  6'. 

If  log  sin  =  9.47760  —  10,  the  nearest  log  sin  in  the  table  is  9.47774  —  10 
(page  36),  and  the  angle  corresponding  to  this  value  is  17°  29'. 

If  log  tan  =  0.76520=  10.76520  — 10,  the  nearest  log  tan  in  the  table  is 
10.76490  — 10  (page  32),  and  the  angle  corresponding  to  this  value  is  80°  15'. 

26.  If  it  is  desired  to  obtain  the  logarithms  of  the  functions  of 
angles  that  contain  seconds,  or  to  obtain  the  value  of  the  angle  in 
degrees,  minutes,  and  seconds,  from  the  logarithms  of  its  functions, 
interpolation  must  be  employed.     Here  it  must  be  remembered  that, 

The  difference  between  two  consecutive  angles  in  the  table  is  60". 

Log  sin  and  log  tan  increase  as  the  angle  increases ;  log  cos  and 
log  cot  diminish  as  the  angle  increases. 

Find  log  tan  70°  46'  8". 

Page  37.     log  tan  70°  46'  =  0.45731. 

The  difference  between  the  mantissas  of  log  tan  70°  46'  and  log  tan  70°  47' 
is  41,  and^V  of  41  =  5. 

As  the  function  is  increasing,  the  5  must  be  added  to  the  figure  in  the  fifth 
place  of  the  mantissa  45731 ;  and 

Therefore  log  tan  70°  46'  8"  =  0.45736. 

Find  log  cos  47°  35' 4". 

Page  48.     log  cos  47°  35'  =  9.82899  —  10. 

The  difference  between  this  mantissa  and  the  mantissas  of  the  next  log  cos 
is  14,  and  -fa  of  14  =  1. 

As  the  function  is  decreasing,  the  1  must  be  subtracted  from  the  figure  in 
the  fifth  place  of  the  mantissa  82899;  and 

Therefore  log  cos  47°  35'  4"  =  9.82898  -  10. 


XIV  LOGARITHMS. 


Find  the  angle  for  which  log  sin  =  9.45359  —  10. 

Page  35.   The  mantissa  of  the  nearest  smaller  log  sin  in  the  table  is  45334. 

The  angle  corresponding  to  this  value  is  1G°  30'. 

The  difference  between  45334  and  the  given  mantissa,  45359,  is  25. 

The  difference  between  45334  and  the  next  following  mantissa,  45377,  is 
43,  and  ff  of  60"  =  35". 

As  the  function  is  increasing,  the  35"  must  be  added  to  1G°  30' ;  and  the 
required  angle  is  16°  30' 35". 

Find  the  angle  for  which  log  cot  =  0.73478. 

Page  32.   The  mantissa  of  the  nearest  smaller  log  cot  in  the  table  is  73415. 

The  angle  corresponding  to  this  value  is  10°  27'. 

The  difference  between  73415  and  the  given  mantissa  is  63. 

The  difference  between  73415  and  next  following  mantissa  is  71,  and 
ff  of  60"  =  53". 

As  the  function  is  decreasing,  the  53"  must  be  subtracted  from  10°  27'; 
and  the  required  angle  is  10°  26'  7". 

27.  If  log  sec  or  log  esc  of  an  angle  is  desired,  it  may  be  found 
from  the  table  by  the  formulas, 

sec  A  = ;    hence,  log  sec  A  =  colog  cos  A. 

cos  A 

esc  A  = ;    hence,  log  esc  A  =  colosr  sin  A. 

sin  A 

Page  31.     log  sec    8°  28'        =  cologcos  8°  28'         =0.00476. 

Page  42.     log  esc  59°  36'  44"  =  colog  sin  59°  36'  44"  =  0.06418. 

28.  If  a  given  angle  is  between  0°  and  1°,  or  between  89°  and  90°  ; 
or,  conversely,  if  a  given  log  sin  or  log  cos  does  not  lie  between  the 
limits  8.23822  and  9.99992  in  the  table;  or,  if  a  given  log  tan  or 
log  cot  does  not  lie  between  the  limits  8.23829  and  11.76171  in  the 
table  ;  then  pages  21-24  of  Table  III.  must  be  used. 

On  page  21,  log  sin  of  angles  between  0°  and  0°  3f,  or  log  cos  of  the 
complementary  angles  between  89°  57'  and  90°,  are  given  to  every 
second;  for  the  angles  between  0°  and  0°3',  log  tan  =  log  sin,  and 
log  cos  =  0.00000  ;  for  the  angles  between  89°  57'  and  90°,  log  cot  = 
log  cos,  and  log  sin  =  0.00000. 

On  pages  22-24,  log  sin,  log  tan,  and  log  cos  of  angles  between  0° 
and  1°,  or  log  cos,  log  cot,  and  log  sin  of  the  complementary  angles 
between  89°  and  90°,  are  given  to  every  10". 

Whenever  log  tan  or  log  cot  is  not  given,  they  may  be  found  by  the 
formulas, 

log  tan  =  colog  cot.  log  cot  =  colog  tan. 

Conversely,  if  a  given  log  tan  or  log  cot  is  not  contained  in  the 
table,  then  the  colog  must  be  found ;  this  will  be  the  log  cot  or 
log  tan,  as  the  case  may  be,  and  will  be  contained  in  the  table. 


INTRODUCTION.  XV 


On  pages  25-27  the  logarithms  of  the  functions  of  angles  between 
1°  and  2°,  or  between  88°  and  90°,  are  given  in  the  manner  employed 
on  pages  22-24.  These  pages  should  be  used  if  the  angle  lies  between 
these  limits,  and  if  not  only  degrees  and  minutes,  but  degrees,  min- 
utes, and  multiples  of  10"  are  given  or  required. 

When  the  angle  is  between  0°  and  2°,  or  88°  and  90°,  and  a  greater 
degree  of  accuracy  is  desired  than  that  given  b}r  the  table,  interpo- 
lation may  be  emploj'ed  ;  but  for  these  angles  interpolation  does  not 
always  give  true  results,  and  it  is  better  to  use  Table  IV. 

Find  log  tan  0°  2'  47",  and  log  cos  89°  37'  20". 

Page  21.       log  tan  0°  2>  47"     =  log  sin  0°  2'  47"  =  6.90820-10. 
Page  23.       log  cos  89°  37'  20"  =  7.81911-10. 

Find  log  cot  0°  2' 15". 

10  -10 

Page  21.       log  tan  0°  2'  15"      =    6.81591  -  10 

Therefore,  log  cot  0°  2' 15"     =    3.18409 

Find  log  tan  89°  38' 30". 

10  -10 

Page  23.       log  cot  89°  38'  30"  =    7.79617  -  10 

Therefore,   log  tan  89°  38'  30"  =    2.20383 

Find  the  angle  for  which  log  tan  =  6.92090  —  10. 

Page  21.     The  nearest  log  tan  is  6.92110  —  10. 
The  corresponding  angle  for  which  is  0°  2'  52". 

Find  the  angle  for  which  log  cos  =  7.70240  —  1 0. 

Page  22.     The  nearest  log  cos  is  7.70261  —  10. 
The  corresponding  angle  for  which  is  89°  42'  40". 

Find  the  angle  for  which  log  cot  =  2.37368. 

This  log  cot  is  not  contained  in  the  table. 
The  cologcot  =  7.62632  —  10  =  log  tan. 

The  log  tan  in  the  table  nearest  to  this  is  (page  22)  7.62510  —  10,  and  the 
angle  corresponding  to  this  value  of  log  tan  is  0°  14' 30". 

29.  If  an  angle  x  is  between  90°  and  360°,  it  follows,  from  formu- 
las established  in  Trigonometiy,  that, 


between  90°  and  180°,  between  180°  and  270c 


log  sin  x  =  log  sin  (180°  —  x) ,  log  sin  x  =  log  sin  (x  —  180°)n, 

log  cos  x  =  log  cos  ( 1 80°  —  x)  n,  log  cos  x  =  log  cos  (x  —  1 80°) „, 

log  tan  x  =  log  tan  ( 1 80°  —  x)  n,  log  tan  x  =  log  tan  (x  —  1 80°) , 

log  cot  x  =  log  cot  ( 180°  —  *)  u  ;  log  cot  x  =  log  cot  (x  —  180°)  ; 


XVI  LOGARITHMS. 


between  270°  and  360°, 
log  sin  x  =  log  sin  (360°  —  x)n, 
log  cos  x  =  log  cos  (360°  —  x) , 
log  tan#  =  log  tan  (360°  —  x)n, 
log  cot  x  =  log  cot  (360°  -  x)n. 
The  letter  n  is  placed  (according  to  custom)  after  the  logarithms 
of  those  functions  which  are  negative  in  value. 

.  The  above  formulas  show,  without  further  explanation,  how  to  find 
by  means  of  Table  III.  the  logarithms  of  the  functions  of  any  angle 
between  90°  and  360°. 

Thus,  log  sin  137°  45'  22"  =  log  sin  42°  14'  38"  =  9.82756  -  10. 
log  cos  137°  45'  22"  =  logn  cos  42°  14'  38"  =  9.86940n  -  10. 
log  tan  137°  45'  22"  =  log„tan  42°  14'  38"  =  9.95815n-  10. 
log  cot  137°  45' 22"  =  logncot  42°  14'  38"  =  0.04185*. 
log  sin  209°  32'  50"  =  lognsin  29°  32'  50"  =  9.69297„  -  10. 
log  cos  330°  27'  10"  =  log  cos    29°  32'  50"  =  9.93949    -  10. 

Conversely,  to  a  given  logarithm  of  a  trigonometric  function  there 
correspond  between  0°  and  360°  four  angles,  one  angle  in  each  quad- 
rant, and  so  related  that  if  x  denote  the  acute  angle,  the  other  three 
angles  are  180°-  x,  180°  +  x,  and  360°-  x. 

If  besides  the  given  logarithm  it  is  known  whether  the  function  is 
positive  or  negative,  the  ambiguity  is  confined  to  two  quadrants, 
therefore  to  two  angles. 

Thus,  if  the  log  tan  =  9.47451  - 10,  the  angles  are  16°  36'  17"  in  Quadrant  I. 
and  196°  36'  17"  in  Quadrant  III. ;  but  if  the  log  tan  =  9.47451„  —  10,  the  angles 
are  163°  23'  43"  in  Quadrant  II.  and  343°  23'  43"  in  Quadrant  IV. 

To  remove  all  ambiguity,  further  conditions  are  required,  or  a 
knowledge  of  the  special  circumstances  connected  with  the  problem  in 
question. 

TABLE    IV. 

30.  This  table  (page  50)  must  be  used  when  great  accuracy  is 
desired  in  working  with  angles  between  0°  and  2°,  or  between  88° 
and  90°. 

The  values  of  S  and  T  are  such  that  when  the  angle  a  is  expressed 
in  seconds, 

S  =  log  sin  a  —  log  a", 

T  =  log  tan  a  —  log  a". 

Hence  follow  the  formulas  given  on  page  50. 

The  values  of  S  and  T  are  printed  with  the  characteristic  10  too 
large,  and  in  using  them  —10  must  always  be  annexed. 


INTRODUCTION.                                                Xvii 

Find  log  sill  0°  58' 17". 

Find  log  cos  88°  26' 41. 2". 

0°  58' 17"  =  3497" 

90° -88°  26' 41.2"  =  1°33'  18.8" 

log  3497  =3.54370 

=  5598.8" 

S  =  4.68555  -10 

log  5598.8  =  3.74809 

S  =  4.68552  -  10 

log  sin  0°  58'  17"  =  8.22925  -  10 

log  cos  88°  26'  41.2"  =  8.43361  - 10 

Find  log  tan  0°  52' 47.5". 

Find  log  tan  89°  54'  37.362". 

0°  52' 47.5"  =  3167.5" 

90°  —  89°  54'  37.362"  =  0°  5'  22.638' 

log  3167.5  =  3.50072 

=  322.638" 

T  =  4.68561  -10 

log  322.638  =  2.50871 

T  =  4.68558  -10 

log  tan  0°  52'  47.5"  =  8. 18633  - 10 

log  cot  89°  54'  37.362"  =  7. 19429  -  10 

log  tan  89°  54'  37.362"  =  2.80571 

Find  the  angle,  if  log  sin  =  6.72306-10. 

6.72306-10 

S  =  4.68557  —  10 

Subtract,     2.03749          =  log  109.015 

109.015"        =  0°1' 49.015" 

Find  the  angle  for  which  log  cot  =  1.67604. 

cologcot  =  8.32396  —  10 

T  =  4.68564  -  10 

Subtract,     3.63832         =  log  4348.3 

4348.3"  =  1°  12'  28.3" 

Find  the  angle  for  which  log  tan  =  1.55407. 

cologtan  =  8.44593  —  10 

T  =  4.68569  -  10 

Subtract,      3.76024          =  log  5757.6 

5757.6"  =  1°  35' 57.6", 

and       90°  -  1°  35'  57.6"  =  88°  24'  2.4". 

Therefore,  the  angle  required  is  88°  24'  2.4". 

TABLE  V. 

31.   This  table  (p.  51),  containing  the  circumferences  and  areas 

of  circles,  does  not  require  explanation. 

TABLE  VI. 

32.   Table  VI.  (pp.  52-69)  contains  the  natural  sines,  cosines, 

tangents,  and  cotangents  of  angles  from  0°  to  90°,  at  intervals  of  lf. 

If  greater  accuracy  is  desired  it  may  be  obtained  by  interpolation. 

Note.    In  preparing  the  preceding  explanations,  we  have  made  free  use 
of  the  Logarithmic  Tables  by  F.  G.  Gauss.     For  Table  VI.  we  are  indebted 
to  D.  Carhart. 

XV111 


LOGARITHMS. 


TABLE    VII. 

33.  This  table  (pp.  70-75)  gives  the  latitude  and  departure  to 
three  places  of  decimals  for  distances  from  1  to  10,  corresponding 
to  bearings  from  0°  to  90°  at  intervals  of  15'. 

If  the  bearing  does  not  exceed  45°  it  is  found  in  the  left-hand 
column,  and  the  designations  of  the  columns  under  "Distance"  are 
taken  from  the  top  of  the  page ;  but  if  the  bearing  exceeds  45°,  it  is 
found  in  the  right-hand  column,  and  the  designations  of  the  columns 
under  "Distance"  are  taken  from  the  bottom  of  the  page. 

The  method  of  using  the  table  will  be  made  plain  by  the  following 
examples :  — 

(1)  Let  it  be  required  to  find  the  latitude  and  departure  of  the 
course  N.  35°  15'  E.  6  chains. 

On  p.  60,  left-hand  column,  look  for  35°  15' ;  opposite  this  bearing,  in  the 
vertical  column  headed  "Distance  6,"  are  found  4.900  and  3.463  under  the 
headings  "Latitude"  and  "Departure"  respectively.  Hence,  latitude  or 
northing  =  4.900  chains,  and  departure  or  easting  as  3.463  chains. 

(2)  Let  it  be  required  to  find  the  latitude  and  departure  of  the 
course  S.  87°  W.  2  chains. 

As  the  bearing  exceeds  45°,  we  look  in  the  right-hand  column  of  p.  55,  and 
opposite  87°  in  the  column  marked  "  Distance  2  "  we  find  (taking  the  designa- 
tions of  the  columns  from  the  bottom  of  the  page)  latitude  =  .105  chains, 
and  departure  =  1.997  chains.  Hence,  latitude  or  southing  =  .105  chains,  and 
departure  or  westing  —  1.997  chains. 

(3)  Let  it  be  required  to  find  the  latitude  and  departure  of  the 
course  N.  15°  45'  W.  27.36  chains. 

In  this  case  we  find  the  required  numbers  for  each  figure  of  the  distance 
separately,  arranging  the  work  as  in  the  following  table.  In  practice,  only 
the  last  columns  under  "  Latitude  "  and  "  Departure  "  are  written. 


Distance. 

Latitude. 

Departure. 

20       =  2  X  10 
7 

0.3    =  3-10 
0.06  =  6-100 

1.925  X  10    =  19.25 
6.737 

2.887  +  10    as    0.289 
5.775 -100=    0.058 

0.543  X  10    =  5.43 
1.90 
0.814-10    =0.081 
1.628  -  100  =  0.016 

27.36 

26.334 

7.427 

Hence,  latitude  =  26.334  chains,  and  departure  =  7.427  chains. 


TABLE   I 

■ 

THE 

COMMON   OR 

BRIGGS 

LOGARITHMS 

, 

OF    THE 

IsTATTJBAL    NUMBEES 

From  1  to  10000. 

1-100 

I 

log 

V 

log 

N 

log 

N 

log 

*T 

log 

1 

0.00  000 

21 

1.  32  222 

41 

1.61278 

61 

1.  78  533 

81 

1. 90  849 

2 

0. 30  103 

22 

1.  34  242 

42 

1.  62  325 

62 

1.  79  239 

82 

1. 91  381 

3 

0.47  712 

23 

1.36173 

43 

1. 63  347 

63 

1.79  934 

83 

1. 91  908 

4 

0.  60  206 

24 

1.38  021 

44 

1.  64  345 

64 

1.80  618 

84 

1. 92  428 

5 

0.69  897 

25 

1.  39  794 

45 

1.65  321 

65 

1.  81  291 

85 

1.  92  942 

6 

0.77  815 

26 

1.41497 

46 

1.  66  276 

66 

1.  81  954 

86 

1.93  450 

7 

0.  84  510 

27 

1.43136 

47 

1.67  210 

67 

1.82  607 

87 

1.93  952 

8 

0.  90  309 

28 

1.44  716 

48 

1.  68  124 

68 

1.83  251 

88 

1.94  448 

9 

0. 95  424 

29 

1.46  240 

49 

1.69  020 

69 

1.83  885 

89 

1.94  939 

10 

1.00000 

30 

1.47  712 

50 

1.  69  897 

70 

1.  84  510 

90 

1.95  424 

11 

1.04139 

31 

1.49136 

51 

1.70  757 

71 

1.85  126 

91 

1.95  904 

12 

1.07  918 

32 

1.50  515 

52 

1.  71  600 

72 

1.  85  733 

92 

1.96  379 

13 

1.11394 

33 

1.51851 

53 

1.72  428 

73 

1.86  332 

93 

1.96  848 

14 

1.14  613 

34 

1.  53  148 

54 

1.  73  239 

74 

1.86  923 

94 

1.97  313 

15 

1.17609 

35 

1.54  407 

55 

1.  74  036 

75 

1.  87  506 

95 

1.  97  772 

16' 

1.20412 

36 

1.55  630 

56 

1.74  819 

76 

1.88  081 

•96 

1. 98  227 

17 

1.  23  045 

37 

1.56  820 

57 

1.  75  587 

77 

1.88  649 

97 

1.98  677 

18 

1.25  527 

38 

1.57  978 

58 

1.76  343 

78 

1. 89  209 

98 

1. 99  123 

19 

1.27  875 

39 

1.  59  106 

59 

1.77  085 

79 

1.  89  763 

99 

1.99  564 

20 

I 

1.30103 

40 

1.60  206 

60 

1.77  815 

80 

1.90  309 

100 

2.00  000 

log 

V 

log 

N 

log 

N 

log 

K 

log 

1-100 


100-150 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100 

00  000 

00  043 

00  0S7 

00130 

00173 

00  217 

00  260 

00  303 

00  346 

00  389 

101 

00  432 

00  475 

00  518 

00  561 

00  604 

00  647 

00  6S9 

00  732 

00  775 

00  817 

102 

00  860 

00  903 

00  945 

00  988 

01030 

01072 

01115 

01157 

01199 

01242 

103 

01284 

01326 

01  368 

01410 

01452 

01494 

01536 

01578 

01620 

01662 

104 

01703 

01745 

01787 

01828 

01870 

01912 

01953 

01995 

02  036 

02  078 

105 

02119 

02160 

02  202 

02  243 

02  284 

02  325 

02  366 

02  407 

02  449 

02  490 

106 

02  531 

02  572 

02  612 

02  653 

02  694 

02  735 

02  776 

02  816 

02  857 

02  898 

107 

02  938 

02  979 

03  019 

03  060 

03  100 

03  141 

03  181 

03  222 

03  262 

03  302 

108 

03  342 

03  383 

03  423 

03  463 

03  503 

03  543 

03  5S3 

03  623 

03  663 

03  703 

109 

03  743 

03  782 

03  822 

03  862 

03  902 

03  941 

03  981 

04  021 

04  060 

04100 

110 

04139 

04179 

04  218 

04  258 

04  297 

04  336 

04  376 

04  415 

04  454 

04  493 

111 

04  532 

04  571 

04  610 

04  650 

04  689 

04  727 

04  766 

04S05 

04  844 

04  883 

112 

04  922 

04  961 

04  999 

05  038 

05  077 

05  115 

05  154 

05  192 

05  231 

05  269 

113 

05  308 

05  346 

05  385 

05  423 

05  461 

05  500 

05  538 

05  576 

05  614 

05  652 

114 

05  690 

05  729 

05  767 

05  805 

05  843 

05  881 

05  918 

05  956 

05  994 

06  032 

115 

06  070 

06108 

06145 

06183 

06  221 

06  258 

06  296 

06  333 

06  371 

06  408 

116 

06  446 

06  483 

06  521 

06  558 

06  595 

06  633 

06  670 

06  707 

06  744 

06  7S1 

117 

06  819 

06  856 

06  893 

06  930 

06  967 

07  004 

07  041 

07  07S 

07115 

07  151 

118 

07188 

07  225 

07  262 

07  298 

07  335 

07  372 

07  408 

07  445 

07  482 

07  518 

119 

07  555 

07  591 

07  628 

07  664 

07  700 

07  737 

07  773 

07S09 

07  846 

07  8S2 

120 

07  918 

07  954 

07  990 

08  027 

08  063 

08  099 

08135 

08171 

08  207 

OS  243 

121 

08  279 

08  314 

08  350 

08  386 

08  422 

OS  458 

08  493 

08  529 

08  565 

08  600 

122 

08  636 

08  672 

OS  707 

08  743 

08  778 

08  814 

OS  849 

08  884 

OS  920 

08  955 

123 

08  991 

09  026 

09  061 

09  096 

09  132 

09167 

09  202 

09  237 

09  272 

09  307 

124 

09  342 

09  377 

09  412 

09  447 

09  482 

09  517 

09  552 

09  587 

09  621 

09  656 

125 

09  691 

09  726 

09  760 

09  795 

09  830 

09  864 

09  899 

09  934 

09  96S 

10  003 

126 

10  037 

10  072 

10106 

10140 

10175 

10  209 

10  243 

10  278 

10  312 

10  346 

127 

10  380 

10  415 

10  449 

10  483 

10  517 

10  551 

10  5S5 

10  619 

10  653 

10  6S7 

128 

10  721 

10  755 

10  789 

10  823 

10  857 

10  890 

10  924 

10  958 

10  992 

11025 

129- 

11059 

11093 

11126 

11160 

11193 

11227 

11261 

11294 

11327 

11361 

130 

11394 

11428 

11461 

11494 

11528 

11561 

11594 

11628 

11661 

11694 

131 

11727 

11760 

11793 

11826 

11860 

11893 

11926 

11959 

11992 

12  024 

132 

12  057 

12  090 

12123 

12  156 

12  189 

12  222 

12  254 

12  287 

12  320 

12  352 

133 

12  385 

12  418 

12  450 

12  483 

12  516 

12  548 

12  581 

12  613 

12  646 

12  67S 

134 

12  710 

12  743 

12  775 

12  808 

12  840 

12  872 

12  905 

12  937 

12  969 

13  001 

135 

13  033 

13  066 

13  098 

13  130 

13  162 

13  194 

13  226 

13  258 

13  290 

13  322 

136 

13  354 

13  386 

13  418 

13  450 

13  481 

13  513 

13  545 

13  577 

13  609 

13  640 

137 

13  672 

13  704 

13  735 

13  767 

13  799 

13  830 

13  862 

13  893 

13  925 

13  956 

138 

13  988 

14  019 

14  051 

14  082 

14  114 

14145 

14176 

14  20S 

14239 

14  270 

139 

14  301 

14  333 

14  364 

14  395 

14  426 

14  457 

14  489 

14  520 

14  551 

14  5S2 

140 

14  613 

14  644 

14  675 

14  706 

14  737 

14  768 

14  799 

14S29 

14  860 

'14  S91 

141 

14  922 

14  953 

14  983 

15  014 

15  045 

15  076 

15  106 

15  137 

15  16S 

15  19S 

142 

15  229 

15  259 

15  290 

15  320 

15  351 

15  381 

15  412 

15  442 

15  473 

15  503 

143 

15  534 

15  564 

15  594 

15  625 

15  655 

15  685 

15  715 

15  746 

15  776 

15S06 

144 

15  836 

15  866 

15  897 

15  927 

15  957 

15  987 

16  017 

16  047 

16  077 

16107 

145 

16137 

16167 

16197 

16  227 

16  256 

16  286 

16  316 

16  346 

16  376 

16  406 

146 

16435 

16  465 

16  495 

16  524 

16  554 

16  584 

16  613 

16  643 

16  673 

16  702 

147 

16  732 

16  761 

16  791 

16  820 

16  850 

16  879 

16  909 

16  938 

16  967 

16  997 

148 

17  026 

17  056 

17  085 

17114 

17143 

17173 

17  202 

17  231 

17  260 

17  2S9 

149 

17  319 

17  348 

17  377 

17  406 

17  435 

17  464 

17  493 

17  522 

17  551 

17  580 

150 

17  609 

17  638 

17  667 

17  696 

17  725 

17  754 

17  782 

17  811 

17  840 

17S69 

N 

0 

1 

o 

3 

4 

5 

G 

7 

8 

9 

100-150 


150-200 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

150 

17  609 

17  638 

17  667 

17  696 

17  725 

17  754 

17  782 

17  811 

17  840 

17  869 

151 

17  898 

17  926 

17  955 

17  984 

18  013 

18  041 

18  070 

18  099 

18127 

18156 

152 

18184 

18  213 

18  241 

18  270 

18  298 

18  327 

18  355 

18  384 

18  412 

18  441 

153 

18  469 

18  498 

18  526 

18  554 

18  583 

18  611 

18  639 

18  667 

18  696 

IS  724 

154 

18  752 

18  780 

18  808 

18  837 

18  865 

18  893 

18  921 

18  949 

18  977 

19  005 

155 

19  033 

19  061 

19  089 

19  117 

19145 

19173 

19  201 

19  229 

19  257 

19  285 

156 

19  312 

19  340 

19  368 

19  396 

19  424 

19  451 

19  479 

19  507 

19  535 

19  562 

157 

19  590 

19  618 

19  645 

19  673 

19  700 

19  728 

19  756 

19  783 

19  811 

19  838 

158 

19  866 

19  893 

19  921 

19  948 

19  976 

20  003 

20  030 

20  058 

20  085 

20112 

159 

20140 

20167 

20194 

20  222 

20  249 

20  276 

20  303 

20  330 

20  358 

20  385 

160 

20412 

20439 

20466 

20493 

20  520 

20  548 

20  575 

20  602 

20  629 

20  656 

161 

20  683 

20  710 

20  737 

20  763 

20  790 

20  817 

20  844 

20  871 

20  898 

20  925 

162 

20  952 

20  978 

21005 

21032 

21059 

21085 

21112 

21139 

21165 

21192 

163 

21219 

21245 

21272 

21299 

21325 

21352 

21378 

21405 

21431 

21458 

164 

21484 

21511 

21537 

21564 

21590 

21617 

21643 

21669 

21696 

21722 

165 

21748 

21  775 

21801 

21827 

21854 

21880 

21906 

21932 

21958 

21985 

166 

22  011 

22  037 

22  063 

22  089 

22  115 

22141 

22  167 

22  194 

22  220 

22  246 

167 

22  272 

22  298 

22  324 

22  350 

22  376 

22  401 

22  427 

22  453 

22  479 

22  505 

168 

22  531 

22  557 

22  583 

22  608 

22  634 

22  660 

22  686 

22  712 

22  737 

22  763 

169 

22  789 

22  814 

22  840 

22  866 

22  891 

22  917 

22  943 

22  968 

22  994 

23  019 

170 

23  045 

23  070 

23  096 

23  121 

23  147 

23172 

23  198 

23  223 

23  249 

23  274 

171 

23  300 

23  325 

23  350 

23  376 

23  401 

23  426 

23  452 

23  477 

23  502 

23  528 

172 

23  553 

23  578 

23  603 

23  629 

23  654 

23  679 

23  704 

23  729 

23  754 

23  779 

173 

23  805 

23  830 

23  855 

23  880 

23  905 

^930 

23  955 

23  980 

24  005 

24  030 

174 

24  055 

24  080 

24105 

24130 

24155 

24180 

24  204 

24  229 

24  254 

24  279 

175 

24  304 

24  329 

24  353 

24  378 

24  403 

24  428 

24  452 

24  477 

24  502 

24  527 

176 

24  551 

24  576 

24  601 

24  625 

24  650 

24  674 

24  699 

24  724 

24  748 

24  773 

177 

24  797 

24  822 

24  846 

24  871 

24  895 

24  920 

24  944 

24  969 

24  993 

25  018 

178 

25  042 

25  066 

25  091 

25  115 

25139 

25164 

25  188 

25  212 

25  237 

25  261 

179 

25  285 

25  310 

25  334 

25  358 

25  382 

25  406 

25  431 

25  455 

25  479 

25  503 

180 

25  527 

25  551 

25  575 

25  600 

25  624 

25  648 

25  672 

25  696 

25  720 

25  744 

181 

25  768 

25  792 

25  816 

25  840 

25  864 

25  888 

25  912 

25  935 

25  959 

25  983 

182 

26  007 

26  031 

26  055 

26  079 

26102 

26126 

26150 

26174 

26198 

26  221 

183 

26  245 

26  269 

26  293 

26  316 

26  340 

26  364 

26  387 

26  411 

26435 

26458 

184 

26482 

26  505 

26  529 

26  553 

26  576 

26  600 

26  623 

26  647 

26  670 

26  694 

185 

26  717 

26  741 

26  764 

26  788 

26  811 

26  834 

26  858 

26  881 

26905 

26  928 

186 

26  951 

26  975 

26  998 

27021 

27  045 

27  068 

27  091 

27114 

27138 

27161 

187 

27184 

27  207 

27  231 

27  254 

27  277 

27  300 

27  323 

27  346 

27  370 

27  393 

188 

27  416 

27  439 

27462 

27485 

27  508 

27  531 

27  554 

27  577 

27  600 

27  623 

189 

27  646 

27  669 

27  692 

27  715 

27  738 

27  761 

27  784 

27  807 

27  830 

27  852 

190 

27  875 

27  898 

27921 

27  944 

27  967 

27  989 

28  012 

28  035 

28058 

2S081 

191 

28103 

28  126 

28149 

28171 

28194 

28  217 

28  240 

28  262 

28  285 

28  307 

192 

28  330 

28  353 

28  375 

28  398 

28  421 

28  443 

28  466 

28  488 

28  511 

28  533 

193 

28  556 

28  578 

28  601 

28  623 

28  646 

28  668 

28  691 

28  713 

28  735 

28  758 

194 

28  780 

28  803 

28  825 

28  847 

28  870 

28  892 

28  914 

28  937 

28  959 

28981 

195 

29003 

29  026 

29  048 

29070 

29092 

29115 

29137 

29159 

291S1 

29  203 

196 

29  226 

29  248 

29  270 

29  292 

29  314 

29  336 

29  358 

29  380 

29  403 

29  425 

197 

29  447 

29  469 

29491 

29  513 

29  535 

29  557 

29  579 

29  601 

29  623 

29  645 

198 

29  667 

29  688 

29  710 

29  732 

29  754 

29  776 

29  798 

29  820 

29  842 

29  863 

199 

29  885 

29  907 

29  929 

29  951 

29  973 

29994 

30  016 

30  03S 

30  060 

30  0S1 

200 

30103 

30125 

30146 

30168 

30  190 

30  211 

30  233 

30  255 

30  276 

30  298 

N 

0 

1 

2 

3 

4 

o 

G 

7 

8 

9 

150  -  200 


200-250 


N 

0     12     3     4 

5     6     7     8 

9 

200 

30103  30125  30146  30168  30190 

30  211  30  233  30  255  30  276 

30  298 

201 

30  320  30  341  30  363  30384  30406 

30428  30  449  30  471  30  492 

30  514 

202 

30  535  30  557  30  578  30  600  30  621 

30  643  30  664  30  685  30  707 

30  728 

203 

30  750  30  771  30  792  30  814  30  835 

30  856  30  878  30  899  30920 

30  942 

204 

30  963  30  984  31006  31027  31048 

31069  31091  31112  31133 

31154 

205 

31175  31197  31218  31239  31260 

31281  31302  31323  31345 

31366 

206 

31387  31408  31429  31450  31471 

31492  31513  31534  31555 

31576 

207 

31597  31618  31639  31660  31681 

31702  31723  31744  31765 

31785 

208 

31806  31827  31848  31869  31890 

31911  31931  31952  31973 

31994 

209 

32  015  32  035  32  056  32  077  32  098 

32118  32  139  32  160  32181 

32  201 

210 

32  222  32  243  32  263  32  284  32  305 

32  325  32  346  32  366  32  387 

32  408 

211 

32  428  32  449  32  469  32  490  32  510 

32  531  32  552  32  572  32  593 

32  613 

212 

32  634  32  654  32  67£  32  695  32  715 

32  736  32  756  32  777  32  797 

32  818 

213 

32  838  32  858  32  879  32  899  32  919 

32  940  32  960  32  980  33  001 

33  021 

214 

33  041  33  062  33  082  33  102  33122 

33143  33  163  33  183  33  203 

33  224 

215 

33  244  33  264  33  284  33  304  33  325 

33  3^5  33  36S  33  385  33  405 

33  425 

216 

33  445  33  465  33  486  33  506  33  526 

33  546  33  566  33  586  33  606 

33  626 

217 

33  646  33  666  33  686  33  706  33  726 

33  746  33  766  33  786  33  806 

33  826 

218 

33  846  33  866  33  885  33  905  33  925 

33  945  33  965  33  985  34  005 

34  025 

219 

34  044  34  064  34  084  34104  34124 

34143  34163  34183  34  203 

34  223 

220 

34  242  34  262  34  282  34  301  34  321 

34  341  34  361  34  380  34  400 

34  420 

221 

34  439  34  459  34  479  34  498  34  518 

34  537  34  557  34  577  34  596 

34  616 

222 

34  635  34  655  34  674  34  694  34  713 

34  733  34  753  34  772  34  792 

34  811 

223 

34  830  34  850  34  869  34  889  34  908 

34  928  34  947  34  967  34  986 

35  005 

224 

35  025  35  044  35  064  35  083  35  102 

35122  35  141  35  160  35  180 

35  199 

225 

35  218  35  238  35  257  35  276  35  295 

35  315  35  334  35  353  35  372 

35  392 

226 

35  411  35  430  35  449  35  468  35  488 

35  507  35  526  35  545  35  564 

35  5S3 

227 

35  603  35  622  35  641  35  660  35  679 

35  698  35  717  35  736  35  755 

35  774 

228 

35  793  35  813  35  832  35  851  35  870 

35  889  35  908  35  927  35  946 

35  965 

229 

35  984  36003  36  021  36  040  36  059 

36  078  36  097  36  116  36135 

36154 

230 

36173  36192  36  211  36  229  36  248 

36  267  36  286  36  305  36  324 

36  342 

231 

36  361  36  380  36  399  36  418  36  436 

36  455  36  474  36  493  36  511 

36  530 

232 

36  549  36  5.68  36  586  36  605  36  624 

36  642  36  661  36  680  36  698 

36  717 

233 

36  736  36  754  36  773  36  791  36  810 

36  829  36  847  36  866  36  884 

36  903 

234 

36  922  36  940  36  959  36  977  36  996 

37  014  37  033  37  051  37  070 

37  088 

235 

37107  37125  37144  37162  37181 

37199  37  218  37  236  37  254 

37  273 

236 

37  291  37  310  37  328  37  346  37  365 

37  383  37  401  37  420  37  438 

37  457 

237 

37475  37  493  37  511  37  530  37  548 

37  566  37  585  37  603  37  621 

37  639 

238 

37  658  37  676  37  694  37  712  37  731 

37  749  37  767  37  785  37  803 

37S22 

239 

37  840  37  858  37  876  37  894  37  912 

37  931  37  949  37  967  37  985 

38  003 

240 

38  021  38  039  38  057  38  075  38  093 

38112  3S130  3S148  3S 166 

3S1S4 

241 

38  202  38  220  38  238  38  256  38  274 

38  292  38  310  38  328  3S346 

38  364 

242 

38  382  38  399  38  417  38  435  38  453 

38  471  38  489  3S  507  38  525 

38  543 

243 

38  561  38  578  38  596  38  614  38  632 

38  650  38  66S  3S6S6  38  703 

38  721 

244 

38  739  38  757  38  775  38  792  38  810  ' 

38  828  38  846  38  863  3S881 

38S99 

245 

38  917  38  934  38  952  38  970  38  987 

39  005  39  023  39  041  39  058 

39  076 

246 

39  094  39111  39129  39146  39164 

39182  39199  39  217  39  235 

39  252 

247 

39  270  39  287  39  305  39  322  39  340 

39  358  39  375  39  393  39  410 

39  428 

248 

39  445  39  463  39  480  39  498  39  515 

39  533  39  550  39  568  39  5S5 

39  602 

249 

39  620  39  637  39  655  39  672  39  690 

39  707  39  724  39  742  39  759 

39  777 

250 

39  794  39  811  39  829  39  846  39  863 

39  881  39S98  39  915  39  933 

39  950 

N 

0     12     3     4 

5     6     7     8 

9 

200-250 


250  -  300 


N 

0 

1 

2     3 

4 

5 

6 

7 

8 

9 

250 

39  794 

39  811 

39  829  39  846 

39  863 

39  881 

39  898 

39  915 

39  933 

39  950 

251 

39  967 

39  985 

40  002  40  019 

40  037 

40  054 

40  071 

40088 

40106 

40123 

252 

40140 

40157 

40175  40192 

40  209 

40  226 

40  243 

40  261 

40  278 

40  295 

253 

40  312 

40  329 

40  346  40  364 

40  381 

40398 

40415 

40  432 

40  449 

40  466 

254 

40  483 

40  500 

40  518  40  535 

40  552 

40  569 

40  586 

40  603 

40  620 

40  637 

255 

40  654 

40  671 

40  688  40  705 

40  722 

40  739 

40  756 

40  773 

40  790 

40  807 

256 

40  824 

40  841 

40  858  40  875 

40  892 

40  909 

40  926 

40  943 

40  960 

40  976 

257 

40  993 

41010 

41027  41044 

41061 

41078 

41095 

41111 

41128 

41145 

258 

41162 

41179 

41196  41212 

41229 

41246 

41263 

41280 

41296 

41313 

259 

41330 

41347 

41363  41380 

41397 

41414 

41430 

41447 

41464 

41481 

260 

41497 

41514 

41531  41547 

41564 

41581 

41597 

41614 

41631 

41647 

261 

41664 

41681 

41697  41714 

41731 

41747 

41764 

41780 

41797 

41814 

262 

41830 

41847 

41863  41880 

41896 

41913 

41929 

41946 

41963 

41979 

263 

41996 

42  012 

42  029  42  045 

42  062 

42  078  42  095 

42111 

42127 

42  144 

264 

42  160 

42177 

42  193  42  210 

42  226 

42  243 

42  259 

42  275 

42  292 

42  308 

265 

42  325 

42  341 

42  357  42  374 

42  390 

42  406 

42  423 

42  439 

42  455 

42  472 

266 

42  488 

42  504 

42  521  42  537 

42  553 

42  570 

42  586 

42  602 

42  619  42  635 

267 

42  651 

42  667 

42  684  42  700 

42  716 

42  732 

42  749 

42  765 

42  781 

42  797 

268 

42  813 

42  830 

42  846  42  862 

42  878 

42  894 

42  911 

42  927 

42  943 

42  959 

269 

42  975 

42  991 

43  008  43  024 

43  040 

43  056 

43  072 

43  088 

43  104 

43  120 

270 

43136 

43152 

43169  43  185 

43  201 

43  217 

43  233 

43  249 

43  265 

43  281 

271 

43  297 

43  313 

43  329  43  345 

43  361 

43  377 

43  393 

43  409 

43  425 

43  441 

272 

43  457 

43  473 

43  489  43  505 

43  521 

43  537 

43  553 

43  569 

43  584 

43  600 

273 

43  616 

43  632 

43  648  43  664 

43  6S0 

43  696 

43  712 

43  727 

43  743 

43  759 

274 

43  775 

43  791 

43  807  43  823 

43  838 

43  854 

43  870 

43  886 

43  902 

43  917 

275 

43  933 

43  949 

43  965  43  981 

43  996 

44  012 

44  028 

44  044 

44  059 

44  075 

276 

44  091 

44107 

44122  44138 

44154 

44170 

44185 

44  201 

44  217 

44  232 

277 

44  248 

14  264 

44  279  44  295 

44  311 

44  326 

44  342 

44  358 

44  373 

44  389 

278 

44  404 

44  420 

44  436  44  451 

44  467 

44  483 

44  498  44  514  44  529  44  545 

279 

44  560 

44  576 

44  592  44  607 

44  623 

44  638 

44  654 

44  669 

44  685 

44  700 

280 

44  716 

44  731 

44  747  44  762 

44  778 

44  793 

44  809 

44  824 

44  840 

44  855 

281 

44  871 

44  886 

44  902  44  917 

44  932 

44  948 

44  963 

44  979 

44  994 

45  010 

282 

45  025 

45  040 

45  056  45  071 

45  086 

45102 

4513  7 

45  133 

45  148 

45  163 

283 

45  179 

45  194 

45  209  45  225 

45  240 

45  255 

45  271 

45  286 

45  301 

45  317 

284 

45  332 

45  347 

45  362  45  378 

45  393 

45  408 

45  423 

45  439 

45  454 

45  469 

285 

45  484 

45  500  45  515  45  530 

45  545 

45  561 

45  576 

45  591 

45  606 

45  621 

286 

45  637 

45  652 

45  667  45  682 

45  697 

45  712 

45  728 

45  743 

45  758 

45  773 

287 

45  788 

45  803 

45  818  45  834 

45  849 

45  864 

45  879 

45  894 

45  909 

45  924 

288 

45  939 

45  954 

45  969  45  984 

46  000 

46  015 

46  030 

46  045 

46060 

46  075 

289 

46  090 

46105 

46120  46135 

46150 

46165 

46180 

46195 

46  210 

46  225 

290 

46  240 

46  255 

46  270  46  285 

46  300 

46  315 

46  330 

46345 

46359 

46374 

291 

46  389 

46404 

46419,  46434 

46  449 

46464 

46479 

46  494 

46  509 

46  523 

292 

46  538 

46  553 

46  568  46  583 

46  598 

46  613 

46  627 

46  642 

46  657 

46  672 

293 

46  687 

46  702 

46  716  46  731 

46  746 

46  761 

46  776 

46  790 

46  805 

46  820 

294 

46  835 

46  850 

46  864  46  879 

46  894 

46909 

46923 

46  938 

46  953 

46967 

295 

46982 

46  997 

47  012  47  026 

47  041 

47  056 

47  070 

47  085 

47100 

47114 

296 

47129 

47144 

47159  47173 

47188 

47  202 

47  217 

47  232 

47  246 

47  261 

297 

47  276 

47  290 

47  305  47  319 

47  334 

47  349 

47  363 

47  378 

47  392 

47  407 

298 

47  422 

47436 

47451  47465 

47  480 

47494 

47  509 

47  524 

47  538 

47  553 

299 

47  567 

47  582 

47  596  47  611 

47625 

47  640 

47  654 

47  669 

47  683 

47  698 

300 

47  712 

47  727 

47  741  47  756 

47  770 

47  784 

47  799 

47  813 

47  828 

47  842 

N 

0 

1 

2     3 

4 

o 

6 

7 

8 

9 

250  -  300 


300-350 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

300 

47  712 

47  727 

47  741 

47  756 

47  770 

47  784 

47  799 

47  813 

47  828 

47  842 

301 

47  857 

47  871 

47  885 

47  900 

47  914 

47  929 

47  943 

47  958 

47  972 

47  986 

302 

48  001 

48  015 

48  029 

48  044 

48  058 

48  073 

48  087 

48101 

48116 

48  130 

303 

48144 

48159 

48173 

48187 

48  202 

48  216 

48  230 

48  244 

48  259 

48  273  . 

304 

48  287 

48  302 

48  316 

48  330 

48  344 

48*359 

48  373 

48  387 

48  401 

48  416 

305 

48  430 

48  444 

48  458 

48  473 

48  487 

48  501 

48  515 

48  530 

48  544 

48  558 

306 

48  572 

48  586 

48  601 

48  615 

48  629 

48  643 

48  657 

48  671 

48  686 

48  700 

307 

48  714 

48  728 

48  742 

48  756 

48  770 

48  785 

48  799 

48  813 

48  827 

48  841 

308 

48  855 

48  869 

48  883 

48  897 

48  911 

48  926 

48  940 

48  954 

48  968 

48  982 

309 

48  996 

49  010 

49  024 

49  038 

49  052 

49  066 

49  080 

49  094 

49108 

49122 

310 

49136 

49150 

49164 

49178 

49192 

49  206 

49  220 

49  234 

49  248 

49  262 

311 

49  276 

49  290 

49  304 

49  318 

49  332 

49  346 

49  360 

49  374 

49  388 

49  402 

312 

49  415 

49  429 

49  443 

49  457 

49  471 

49  485 

49  499 

49  513 

49  527 

49  541 

313 

49  554 

49  568 

49  582 

49  596 

49  610 

49  624 

49  638 

49  651 

49  665 

49  679 

314 

49  693 

49  707 

49  721 

49  734 

49  748 

49  762 

49  776 

49  790 

49  803 

49  817 

315 

49  831 

49  845 

49  859 

49  872 

49  886 

49  90D 

49  914 

49  927 

49  941 

49  95£ 

316 

49  969 

49  982 

49  996 

50  010 

50  024 

50  037 

50  051 

50  065 

50  079 

50  092 

317 

50106 

50120 

50  133 

50147 

50161 

50174 

50  188" 

50  202 

50  215 

50  229 

318 

50  243 

50  256 

50  270 

50  284 

50  297 

50  311 

50  325 

50  338 

50  352 

50  365 

319 

50  379 

50  393 

50  406 

50  420 

50  433 

50447 

50  461 

50474 

50488 

50  501 

320 

50  515 

50  529 

50  542 

50  556 

50  569 

50  583 

50  596 

50  610 

50  623 

50  637 

321 

50  651 

50  664 

50  678 

50  691 

50  705 

50  718 

50  732 

50  745 

50  759 

50  772 

322 

50  786 

50  799 

50  813 

50  826 

50  840 

50  853 

50  866 

50  880 

50  893 

50  907 

323 

50  920 

50  934 

50  947 

50  961 

50  974 

50  987 

51001 

51014 

51028 

51041 

324 

51055 

51  068 

510S1 

51095 

51108 

51121 

51135 

51148 

51162 

51175 

325 

51188 

51202 

51215 

51228 

51242 

51255 

51268 

51282 

51295 

51  308 

326 

51322 

51335 

51348 

51362 

51375 

51388 

51402 

51415 

51428 

51441 

327 

51455 

51468 

51481 

51495 

51508 

51521 

51534 

51548 

51561 

51574 

328 

51587 

51601 

51614 

51627 

51640 

51654 

51667 

51680 

51693 

51706 

329 

51720 

51733 

51746 

51759 

51772 

51786 

51799 

51812 

51825 

51838 

330 

51851 

51865 

51878 

51891 

51904 

51917 

51  930  51  943 

51  957 

51970 

331 

51983 

51996 

52  009 

52  022 

52  035 

52  048 

52  061 

52  075 

52  088 

52  101 

332 

52114 

52  127 

52  140 

52153 

52  166 

52179 

52  192 

52  205 

52  218 

52  231 

333 

52  244 

52  257 

52  270 

52  284 

52  297 

52  310 

52  323 

52  336 

52  349 

52  362 

334 

52  375 

52  388 

52  401 

52  414 

52  427 

52  440 

52  453 

52  466 

52  479 

52  492 

335 

52  504 

52  517 

52  530 

52  543 

52  556 

52  569 

52  582 

52  595 

52  608 

52  621 

336 

52  634 

52  647 

52  660 

52  673 

52  686 

52  699 

52  711 

52  724 

52  737 

52  750 

337 

52  763 

52  776 

52  789 

52  802 

52  815 

52  827 

52  840 

52  853 

52  866 

52  879 

338 

52  892 

52  905 

52  917 

52  930 

52  943 

52  956 

52  969 

52  982 

52  994 

53  007 

339 

53  020 

53  033 

53  046 

53  058 

53  071 

53  084 

53  097 

53110 

53122 

53  135 

340 

53148 

53  161 

53  173 

53  186 

53  199 

53  212 

53  224 

53  237 

53  250 

53  263 

341 

53  275 

53  288 

53  301 

53  314 

53  326 

53  339 

53  352 

53  364 

53  377 

53  390 

342 

53  403 

53  415 

53  428 

53  441 

53  453 

53  466 

53  479 

53  491 

53  504 

53  517 

343 

53  529 

53  542 

53  555 

53  567 

53  580 

53  593 

53  605 

53  618 

53  631 

53  643 

344 

53  656 

53  668 

53  681 

53  694 

53  706 

53  719 

53  732 

53  744 

53  757 

53  769 

345 

53  782 

53  794 

53  807 

53  820 

53  832 

53  845 

53  857 

53  870 

53  882 

53  895 

346 

53  908 

53  920 

53  933 

53  945 

53  958 

53  970 

53  983 

53  995 

54  008 

54  020 

347 

54  033 

54  045 

54  058 

54  070 

54  083 

54  095 

54108 

54120 

54  133 

54  145 

348 

54158 

54170 

54183 

54195 

54  208 

54  220 

54  233 

54  245 

54  258 

54  270 

349 

54  283 

54  295 

54  307 

54  320 

54  332 

54  345 

54  357 

54  370 

54  382 

54  394 

350 

54  407 

54  419 

54  432 

54  444 

54  456 

54  469 

54  481 

54  494 

54  506 

54  518 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

300-350 


350-400 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

350 

54  407 

54  419 

54  432 

54  444 

54  456 

54  469 

54  481 

54  494 

54  506 

54  518 

351 

54  531 

54  543 

54  555 

54  568 

54  5S0 

54  593 

54  605 

54  617 

54  630 

54  642 

352 

54  654 

54  667 

54  679 

54  691 

54  704 

54  716 

54  728 

54  741 

54  753 

54  765 

353 

54  777 

54  790 

54  802 

54  814 

54  827 

54  839 

54  851 

54  864 

54  876 

54  888 

354 

54  900 

54  913 

54  925 

54  937 

54  949 

54  962 

54  974 

54  986 

54  998 

55  011 

355 

55  023 

55  035 

55  047 

55  060 

55  072 

55  084 

55  096 

55  108 

55  121 

55133 

356 

55  145 

55  157 

55  169 

55  182 

55  194 

55  206 

55  218 

55  230 

55  242 

55  255 

357 

55  267 

55  279 

55  291 

55  303 

55  315 

55  328 

55  340 

55  352 

55  364 

55  376 

358 

55  388 

55  400 

55  413 

55  425 

55  437 

55  449 

55  461 

55  473 

55  485 

55  497 

359 

55  509 

55  522 

55  534 

55  546 

55  558 

55  570 

55  582 

55  594 

55  606 

55  618 

360 

55  630 

55  642 

55  654 

55  666 

55  678 

55  691 

55  703 

55  715 

55  727 

55  739 

361 

55  751 

55  763 

55  775 

55  787 

55  799 

55  811 

55  823 

55  835 

55  847 

55  859 

362 

55  871 

55  883 

55  895 

55  907 

55  919 

55  931 

55  943 

55  955 

55  967 

55  979 

363 

55  991 

56003 

56  015 

56  027 

56  038 

56  050 

56  062 

56  074 

56  086 

56  098 

364 

56110 

56122 

56134 

56146 

56158 

56170 

56182 

56194 

56  205 

56  217 

365 

56  229 

56  241 

56  253 

56  265 

56  277 

56  289 

56  301 

56  312 

56  324 

56  336 

366 

56  348 

56  360 

56  372 

56  384 

56  396 

56  407 

56419 

56431 

56  443 

56455 

367 

56467 

56  478 

56  490 

56  502 

56  514 

56  526 

56  538 

56  549 

56  561 

56  573 

368 

56  585 

56  597 

56  60S 

56  620 

56  632 

56  644 

56  656 

56  667 

56  679 

56  691 

369 

56  703 

56  714 

56  726 

56  738 

56  750 

56  761 

56  773 

56  785 

56  797 

56  808 

370 

56  820 

56  832 

56  844 

56  855 

56  867 

56  879 

56  891 

56  902 

56  914 

56926 

371 

56  937 

56  949 

56  961 

56  972 

56  984 

56  996 

57  008 

57  019 

57  031 

57  043 

372 

57  054 

57  066 

57  078 

57  089 

57101 

57113 

57124 

57136 

57148 

57159 

373 

57171 

57183 

57  194 

57  206 

57  217 

57  229 

57  241 

57  252 

57  264 

57  276 

374 

57  287 

57  299 

57  310 

57  322 

57  334 

57  345 

57357 

57  368 

57  380 

57392 

375 

57  403 

57  415 

57  426 

57  438 

57449 

57461 

57473 

57484 

57496 

57  507 

376 

57  519 

57  530 

57  542 

57  553 

57  565 

57  576 

57  588 

57  600 

57  611 

57  623 

377 

57  634 

57  646 

57  657 

57  669 

57  680 

57  692 

57  703 

57  715 

57  726 

57  738 

378 

57  749 

57  761 

57  772 

57  784 

57  795 

57  807 

57  818 

57  830 

57  841 

57  852 

379 

57  864 

57  875 

57  887 

57  898 

57  910 

57921 

57  933 

57944 

57955 

57967 

380 

57  978 

57  990 

58  001 

58  013 

58  024 

58  035 

58  047 

58  058 

58  070 

58  081 

381 

58  092 

58104 

58115 

58127 

58138 

58149 

58161 

58172 

58184 

58195 

382 

58  206 

58  218 

58  229 

58  240 

58  252 

58  263 

58  274 

58  286 

58  297 

58  309 

383 

58  320 

58  331 

58  343 

58  354 

58  365 

58  377 

58  388 

58  399 

58410 

58422 

384 

58  433 

58  444 

58  456 

58  467 

58  478 

58  490 

58  501 

58  512 

58  524 

58  535 

385 

58  546 

58  557 

58  569 

58  580 

58  591 

58  602 

58  614 

58  625 

58  636 

58  647 

386 

58  659 

58  670 

58  681 

58  692 

58  704 

58  715 

58  726 

58  737 

58  749 

58  760 

387 

58  771 

58  782 

58  794 

58  805 

58  816 

58  827 

58  838 

58  850 

58  861 

58  872 

388 

58  883 

58  894 

58  906 

58  917 

58  928 

58  939 

58950 

58  961 

58  973 

58  984 

389 

58  995 

59  006 

59  017 

59  028 

59  040 

59  051 

59  062 

59  073 

59  084 

59  095 

390 

59106 

59  118 

59129 

59140 

59151 

59162 

59173 

59184 

59195 

59  207 

391 

59  218 

59  229 

59  240 

59  251 

59  262 

59  273 

59  284 

59  295 

59  306 

59  318 

392 

59  329 

59  340 

59  351 

59  362 

59  373 

59  384 

59  395 

59406 

59417 

59428 

393 

59  439 

59450 

59  461' 

59  472 

59  483 

59494 

59  506 

59  517 

59  528 

59  539 

394 

59  550 

59  561 

59  572 

59  583 

59  594 

59  605 

59  616 

59  627 

59  638 

59  649 

395 

59  660 

59  671 

59  682 

59  693 

59  704 

59  715 

59  726 

59  737 

59  748 

59  759 

396 

59  770 

59  780 

59  791 

59  802 

59  813 

59  824 

59  835 

59  846 

59  857 

59  86S 

397 

59  879 

59  890 

59  901 

59  912 

59  923 

59  934 

59945 

59  956 

59  966 

59  977 

398 

59  988 

59  999 

60  010 

60  021 

60  032 

60  043 

60054 

60  065 

60  076 

60  086 

399 

60  097 

60108 

60119 

60130 

60141 

60152 

60163 

60173 

60184 

60195 

400 

60  206 

60  217 

60  228 

60  239 

60  249 

60  260 

60  271 

60  282 

60  293 

60  304 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

350-400 


400-450 


N 

0 

1 

2 

3 

4 

o 

0 

7 

8 

9 

400 

60  206 

60  217 

60  228 

60  239 

60  249 

60  260 

60  271 

60  282 

60  293 

60  304 

401 

60  314 

60  325 

60  336 

60  347 

60  358 

60  369 

60  379 

60  390 

60  401 

60  412 

402 

60  423 

60  433 

60  444 

60  455 

60  466 

60  477 

60487 

60  498 

60  509 

60  520 

403 

60  531 

60  541 

60  552 

60  563 

60  574 

60  584 

60  595 

60  606 

60  617 

60  627 

404 

60  638 

60  649 

60  660 

60  670 

60  681 

60  692 

60  703 

60  713 

60  724 

60  735 

405 

60  746 

60  756 

60  767 

60  778 

60  788 

60  799 

60  810 

60  821 

60  831 

60  842 

406 

60  853 

60  863 

60  874 

60  885 

60  895 

60  906 

60  917 

60  927 

60  938 

60  949 

407 

60  959 

60  970 

60  981 

60  991 

61002 

61013 

61023 

61034 

61045 

61055 

408 

61066 

61077 

61087 

61098 

61109 

61119 

61130 

61140 

61151 

61162 

409 

61172 

61183 

61194 

61204 

61215 

61  225 

61236 

61247 

61257 

61268 

410 

61278 

61289 

61300 

61310 

61321 

61331 

61342 

61352 

61363 

61374 

411 

61  384 

61395 

61405 

61416 

61426 

61437 

61448 

61458 

61469 

61479 

412 

61490 

61500 

61511 

61521 

61532 

61542 

61553 

61563 

61574 

61584 

413 

61595 

61606 

61616 

61627 

61637 

61648 

61658 

61669 

61679 

61690 

414 

61700 

61711 

61721 

61731 

61742 

61752 

61763 

61773 

61784 

61794 

415 

'6/805 

61815 

61826 

61836 

61847 

61857 

61868 

61878 

61888 

61899 

416 

61909 

61920 

61930 

61941 

61  951 

61962 

61972 

61982 

61993 

62  003 

417 

62  014 

62  024 

62  034 

62  045 

62  055 

62  066 

62  076 

62  086 

62  097 

62  107 

418 

62118 

62128 

62138 

62  149 

62  159 

62170 

62  180 

62  190 

62  201 

62  211 

419 

62  221 

62  232 

62  242 

62  252 

62  263 

62  273 

62  284 

62  294 

62  304 

62  315 

420 

62  325 

62  335 

62  346 

62  356 

62  366 

62  377 

62  387 

62  397 

62  408 

62  418 

421 

62  428 

62  439 

62  449 

62  459 

62  469 

62  480 

62  490 

62  500 

62  511 

62  521 

422 

62  531 

62  542 

62  552 

62  562 

62  572 

62  583 

62  593 

62  603 

62  613 

62  624 

423 

62  634 

62  644 

62  655 

62  665 

62  675 

62  685 

62  696 

62  706 

62  716 

62  726 

424 

62  737 

62  747 

62  757 

62  767 

62  778 

62  788 

62  798 

62  808 

62  818 

62  829 

425 

62  839 

62  849 

62  859 

62  870 

62  880 

62  890 

62  900 

62  910 

62  921 

62  931 

426 

62  941 

62  951 

62  961 

62  972 

62  982 

62  992 

63  002 

63  012 

63  022 

63  033 

427 

63  043 

63  053 

63  063 

63  073 

63  083 

63  094 

63  104 

63  114 

63124 

63134 

428 

63  144 

63  155 

63  165 

63  175 

63  185 

63  195 

63  205 

63  215 

63  225 

63  236 

429 

63  246 

63  256 

63  266 

63  276 

63  286 

63  296 

63  306 

63  317 

63  327 

63  337 

430 

63  347 

63  357 

63  367 

63  377 

63  387 

63  397 

63  407 

63  417 

63  428 

63  438 

431 

63  448 

63  458 

63  468 

63  478 

63  488 

63  498 

63  508 

63  518 

63  528 

63  538 

432 

63  548 

63  558 

63  568 

63  579 

63  589 

63  599 

63  609 

63  619 

63  629 

63  639 

433 

63  649 

63  659 

63  669 

63  679 

63  689 

63  699 

63  709 

63  719 

63  729 

63  739 

434 

63  749 

63  759 

63  769 

63  779 

63  789 

63  799 

63  809 

63  819 

63  829 

63  839 

435 

63  849 

63  859 

63  869 

63  879 

63  889 

63  899 

63  909 

63  919 

63  929 

63  939 

436 

63  949 

63  959 

63  969 

63  979 

63  988 

63  998 

64  008 

64  018 

64  028 

64  038 

437 

64  048 

64  058 

64  068 

64  078 

64  088 

64  098 

64108 

64118 

64128 

64137 

438 

64147 

64157 

64167 

64177 

64187 

64197 

64  207 

64  217 

64  227 

64  237 

439 

64  246 

64  256 

64  266 

64  276 

64  286 

64  296 

64  306 

64  316 

64  326 

64  335 

440 

64  345 

64  355 

64  365 

64  375 

64  385 

64  395 

64  404 

64  414 

64  424 

64  434 

441 

64  444 

64454 

64  464 

64  473 

64  483 

64  493 

64  503 

64  513 

64  523 

64  532 

442 

64  542 

64  552 

64  562 

64  572 

64  582 

64  591 

64  601 

64  611 

64  621 

64  631 

443 

64  640 

64  650 

64  660 

64  670 

64  680 

64  689 

64  699 

64  709 

64  719 

64  729 

444 

64  738 

64  748 

64  758 

64  768 

64  777 

64  787 

64  797 

64  807 

64  816 

64  826 

445 

64  836 

64  846 

64  856 

64  865 

64  875 

64  885 

64  895 

64  904 

64  914 

64  924 

446 

64  933 

64  943 

64  953 

64  963 

64  972 

64  982 

64  992 

65  002 

65  011 

65  021 

447 

65  031 

65  040 

65  050 

65  060 

65  070 

65  079 

65  089 

65  099 

65  108 

65  118 

448 

65  128 

65  137 

65147 

65  157 

65  167 

65  176 

65  186 

65  1% 

65  205 

65  215 

449 

65  225 

65  234 

65  244 

65  254 

65  263 

65  273 

65  283 

65  292 

65  302 

65  312 

450 

65  321 

65  331 

65  341 

65  350 

65  360 

65  369 

65  379 

65  389 

65  398 

65  408. 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

400-450 


450  -  500 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

450 

65  321 

65  331 

65  341 

65  350 

65  360 

65  369 

65  379 

65  389 

65  398 

65  408 

451 

65  418 

65  427 

65  437 

65  447 

65  456 

65  466 

65  475 

65  485 

65  495 

65  504 

452 

65  514 

65  523 

65  533 

65  543 

65  552 

65  562 

65  571 

65  581 

65  591 

65  600 

453 

65  610 

65  619 

65  629 

65  639 

65  648 

65  658 

65  667 

65  677 

65  686 

65  696 

454 

65  706 

65  715 

65  725 

65  734 

65  744 

65  753 

65  763 

65  772 

65  782 

65  792 

455 

65  801 

65  811 

65  820 

65  830 

65  839 

65  849 

65  858 

65  868 

65  877 

65  887 

456 

65  896 

65  906 

65  916 

65  925 

65  935 

65  944 

65  954 

65  963 

65  973 

65  982 

457 

65  992 

66  001 

66  011 

66  020 

66  030 

66  039 

66  049 

66  058 

66  068 

66  077 

458 

66087 

66  096 

66106 

66115 

66124 

66134 

66143 

66153 

66162 

66172 

459 

66181 

66191 

66  200 

66  210 

66  219 

66  229 

66  238 

66  247 

66  257 

66  266 

460 

66  276 

66  285 

66  295 

66304 

66  314 

66  323 

66  332 

66  342 

66  351 

66  361 

461 

66  370 

66  380 

66  389 

66  398 

66  408 

66  417 

66427 

66  436 

66  445 

66455 

462 

66  464 

66474 

66483 

66  492 

66  502 

66  511 

66  521 

66  530 

66  539 

66  549 

463 

66  558 

66  567 

66  577 

66  586 

66  596 

66  605 

66  614 

66  624 

66  633 

66  642 

464 

66  652 

66  661 

66  671 

66  680 

66  689 

66  699 

66  708 

66  717 

66  727 

66  736 

465 

66  745 

66  755 

66  764 

66  773 

66  783 

66  792 

66  801 

66  811 

66  820 

66  829 

466 

66  839 

66  848 

66  857 

66  867 

66  876 

66  885 

66  894 

66  904 

66  913 

66922 

467 

66  932 

66941 

66950 

66  960 

66  969 

66  978 

66  987 

66  997 

67  006 

67  015 

468 

67  025 

67  034 

67  043 

67  052 

67  062 

67  071 

67  080 

67  089 

67  099 

67108 

469 

67117 

67127 

67136 

67145 

67154 

67164 

67173 

67182 

67191 

67  201 

470 

67  210 

67  219 

67  228 

67  237 

67  247 

67  256 

67  265 

67  274 

67  284 

67  293 

471 

67  302 

67  311 

67  321 

67  330 

67  339 

67  348 

67  357 

67  367 

67  376 

67  385 

472 

67  394 

67  403 

67  413 

67  422 

67  431 

67  440 

67  449 

67  459 

67  468 

67  477 

473 

67  486 

67  495 

67  504 

67  514 

67  523 

67  532 

67  541 

67  550 

67  560 

67  569 

474 

67  578 

67  587 

67  596 

67  605 

67  614 

67  624 

67  633 

67  642 

67  6*51 

67  660 

475 

67  669 

67  679 

67  688 

67  697 

67  706 

67  715 

67  724 

67  733 

67  742 

67  752 

476 

67  761 

67  770 

67  779 

67  788 

67  797 

67  806 

67  815 

67  825 

67  834 

67  843 

477 

67  852 

67  861 

67  870 

67  879 

67  8S8 

67  897 

67  906 

67  916 

67  925 

67  934 

478 

67  943 

67  952 

67  961 

67  970 

67  979 

67  988 

67  997 

68  006 

68  015 

68  024 

479 

68  034 

68  043 

68  052 

68  061 

68  070 

68  079 

68  088 

68  097 

68106 

68115 

480 

68124 

68133 

68142 

68151 

68160 

68169 

68178 

68187 

68196 

68  205 

481 

68  215 

68  224 

68  233 

68  242 

68  251 

68  260 

68  269 

68  278 

68  287 

68  296 

482 

68  305 

68  314 

68  323 

68  332 

68  341' 

68  350 

68  359 

68  368 

68  377 

68  386 

483 

68  395 

68  404 

68  413 

68  422 

68  431 

68  440 

68  449 

68  458 

68  467 

68  476 

484 

68  485 

68  494 

68  502 

68  511 

68  520 

68  529 

68  538 

68  547 

68  556 

68  565 

485 

68  574 

68  583 

68  592 

68  601 

68  610 

68  619 

68  628 

68  637 

68  646 

68  655 

486 

68  664 

68  673 

68  681 

68  690 

68  699 

68  708 

68  717 

68  726 

68  735 

68  744 

487 

68  753 

68  762 

68  771 

68  780 

68  789 

68  797 

68  806 

68  815 

68  824 

68  833 

488 

68  842 

68  851 

68  860 

68  869 

68  878 

68  886 

68  895 

68  904 

68  913 

68  922 

489 

68  931 

68  940 

68  949 

68  958 

68  966 

68  975 

68  984 

68  993 

69  002 

69^011 

490 

69  020 

69  028 

69  037 

69  046 

69  055 

69  064 

69  073 

69  082 

69  090 

69  099 

491 

69108 

69117 

69126 

69135 

69144 

69152 

69161 

69170 

69179 

69188 

492 

69197 

69  205 

69  214 

69  223 

69  232 

69  241 

69  249 

69  258 

69  267 

69  276 

493 

69  285 

69  294 

69  302 

69  311 

69  320 

69  329 

69  338 

69  346 

69  355 

69  364 

494 

69  373 

69  381 

69  390 

69  399 

69  408 

69  417 

69  425 

69  434 

69  443 

69  452 

495 

69  461 

69  469 

69  478 

69  487 

69  496 

69  504 

69  513 

69  522 

69  531 

69  539 

496 

69  548 

69  557 

69  566 

69  574 

69  583 

69  592 

69  601 

69  609 

69  618 

69  627 

497 

69  636 

69  644 

69  653 

69  662 

69  671 

69  679 

69  688 

69  697 

69  705 

69  714 

498 

69  723 

69  732 

69  740 

69  749 

69  758 

69  767 

69  775 

69  784 

69  793 

69  801 

499 

69  810 

69  819 

69  827 

69  836 

69  84£ 

69  854 

69  862 

69  871 

69  880 

69  888 

500 

69  897 

69  906 

69914 

69  923 

69  932 

69  940 

69  949 

69  958 

69  966 

69  975 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

450  -  500 


10 


500-550 


N 

0 

1 

2     3 

4 

5 

0 

7 

8 

9 

500 

69  897 

69  906 

69  914  69  923 

69  932 

69  940 

69  949 

69  958 

69  966 

69  975 

501 

69  984 

69  992 

70  001  70010 

70  018 

70  027 

70  036 

70  044 

70  053 

70  062 

502 

70  070 

70  079 

70  088  70  096 

70105 

70114 

70122 

70131 

70140 

70148 

503 

70157 

70165 

70174  70183 

70191 

70  200 

70  209 

70  217 

70  226 

70  234 

504 

70  243 

70  252 

70  260  70  269 

70  278 

70  286 

70  295 

70  303 

70  312 

70  321 

505 

70  329 

70  338 

70  346  70  355 

70  364 

70  372 

70  381 

70  389 

70  398 

70  406 

50$ 

70415 

70424 

70  432  70441 

70  449 

70  458 

70  467 

70  475 

70  484 

70492 

507 

70  501 

70  509 

70  518  70  526 

70  535 

70  544 

70  552 

70  561 

70  569 

70  578 

508 

70  586 

70  595 

70  603  70  612 

70  621 

70  629 

70  638 

70  646 

70  65i 

70  663 

509 

70  672 

70  680 

70  689  70  697 

70  706 

70  714 

70  723 

70  731 

70  740 

70  749 

510 

70  757 

70  766 

70  774  70  783 

70  791 

70  800 

70  808 

70  817 

70  825 

70  834 

511 

70  842 

70  851 

70  859  70  868 

70  876 

70  885 

70  893 

70  902 

70  910 

70  919 

512 

70  927 

70  935 

70  944  70  952 

70  961 

70  969 

70  978 

70  986 

70995 

71003 

513 

71012 

71020 

71029  71037 

71046 

71054 

71063 

71071 

71079 

71088 

514 

71096 

71105 

71 113  71  122 

71130 

71139 

71147 

71155 

71164 

71172 

515 

71181 

71189 

71198  71206 

71214 

71223 

71231 

71240 

71248 

71  257 

516 

71265 

71273 

71282  71290 

71299 

71307 

71315 

71324 

71332 

71341 

517 

71349 

71357 

71366  71374 

71383 

71391 

71399 

71408 

71416 

71425 

518 

71433 

71441 

71450  71458 

71466 

71475 

71483 

71492 

71500 

71508 

519 

71517 

71525 

71533  71542 

71550 

71559 

71567 

71575 

71584 

71592 

520 

71600 

71609 

71617  71625 

71634 

71642 

71650 

71659 

71667 

71675 

521 

71684 

71692 

71700  71709 

71717 

71725 

71734 

71742 

71750 

71759 

522 

71767 

71775 

71784  71792 

71800 

71809 

71817 

71825 

71834 

71842 

523 

71850 

71858 

71867  71875 

71883 

71892 

71900 

71908 

71917 

71925 

524 

71933 

71941 

71950  71958 

71966 

71975 

71983 

71991 

71999 

72  008 

525 

72  016 

72  024 

72  032  72  041 

72  049 

72  057 

72  066 

72  074 

72  082 

72  090 

526 

72  099 

72  107 

72  115  72  123 

72132 

72  140 

72148 

72156 

72  165 

72  173 

527 

72181 

72189 

72198  72  206 

72  214 

72  222 

72  230 

72  239 

72  247 

72  255 

528 

72  263 

72  272 

72  280  72  2SS 

72  296 

72  304 

72  313 

72  321 

72  329 

72  337 

529 

72  346 

72  354 

72  362  72  370 

72  378 

72  387 

72  395 

72  403 

72  411 

72419 

530 

72  428 

72  436 

72  444  72  452 

72  460 

72  469 

72  477 

72  485 

72  493 

72  501 

531 

72  509 

72  518 

72  526  72  534 

72  542 

72  550 

72  558 

72  567 

72  575 

72  5S3 

532 

72  591 

72  599 

72  607  72  616 

72  624 

72  632 

72  640 

72  648 

72  656 

72  665 

533 

72  673 

72  681 

72  689  72  697 

72  705 

72  713 

72  722 

72  730 

72  738 

72  746 

534 

72  754 

72  762 

72  770  72  779 

72  787 

72  795 

72  803 

72  811 

72  819 

72  827 

535 

72  835 

72  843 

72  852  72  860 

72  868 

72  876 

72  884 

72  892 

72  900 

72  908 

536 

72  916 

72  925 

72  933  72  941 

72  949 

72  957 

72  965 

72  973 

72  981 

72  989 

537 

72  997 

73  006 

73  014  73  022 

73  030 

73  038 

73  046 

73  054 

73  062 

73  070 

538 

73  07S 

73  086 

73  094  73102 

73111 

73  119 

73  127 

73  135 

73143 

73  151 

539 

73  159 

73  167 

73  175  73  183 

73  191 

73199 

73  207 

73  215 

73  223 

73  231 

540 

73  239 

73  247 

73  255  73  263 

73  272 

73  280 

73  288 

73  296 

73  304 

73  312 

541 

73  320 

73  328 

73  336  73  344 

73  352 

73  360 

73  368 

73  376 

73  384 

73  392 

542 

73  400 

73  408 

73  416  73  424 

73  432 

73  440 

73  448 

73  456 

73  464 

73  472 

543 

73  480 

73  488 

73  496  73  504 

73  512 

73  520 

73  528 

73  536 

73  544 

73  552 

544 

73  560 

73  568 

73  576  73  584 

73  592 

73  600 

73  608 

73  616 

73  624 

73  632 

545 

73  640 

73  648 

73  656  73  664 

73  672 

73  679 

73  687 

73  695 

73  703 

73  711 

546 

73  719 

73  727 

73  735  73  743 

73  751 

73  759 

73  767 

73  775 

73  783 

73  791 

547 

73  799 

73  807 

73  815  73  823 

73  830 

73  838 

73  846 

73  854 

73  862 

73  870 

548 

73  878 

73  886 

73  894  73  902 

73  910 

73  91S 

73  926 

73  933 

73  941 

73  949 

549 

73  957 

73  965 

73  973  73  9S1 

73  989 

73  997 

74  005 

74  013 

74  020 

74  028 

550 

74  036 

74  044 

74  052  74  060 

74  068 

74  076 

74  084 

74  092 

74  099 

74107 

N 

0 

1 

2     3 

4 

5 

6 

7 

8 

9 

500-550 


550  -  600 

11 

N 

0 

1 

2 

3 

4 

5     6 

7 

8 

9 

550 

74  036 

74  044 

74  052 

74  060 

74  068 

74  076  74  0S4 

74  092 

74  099 

74107 

551 

74115 

74123 

74131 

74139 

74147 

74155  74162 

74  170 

74178 

74186 

552 

74  194 

74  202 

74  210 

74  218 

74  225 

74  233  74  241 

74  249 

74  257 

74  265 

553 

74  273 

74  280 

74  288 

74  296 

74  304 

74  312  74  320 

74  327 

7+335 

74  343 

554 

74  351 

74  359 

74  367 

74  374 

74  382 

74  390  74  398 

74  406 

74414 

74  421 

555 

74  429 

74  437 

74  445 

74  453 

74  461 

74  468  74  476 

74  484 

74  492 

74  500 

556 

74  507 

74  515 

74  523 

74  531 

74  539 

74  547  74  554 

74  562 

74  570 

74  57S 

557 

74  586 

74  593 

74  601 

74  609 

74  617 

74  624  74  632 

74  640 

74  648 

74  656 

558 

74  663 

74  671 

74  679 

74  687 

74  695 

74  702  74  710 

74  718 

74  726 

74  733 

559 

74  741 

74  749 

74  757 

74  764 

74  772 

74  780  74  788 

74  796 

74  803 

74  811 

560 

74  819 

74  827 

74  834 

74  842 

74  850 

74  858  74  865 

74  873 

74  881 

74  889 

561 

74  896 

74  904 

74  912 

74  920 

74  927 

74  935  74  943 

74  950 

74  95S 

74  966 

562 

74  974 

74  981 

74  989 

74  997 

75  005 

75  012  75  020 

75  028 

75  035 

75  043 

563 

75  051 

75  059 

75  066 

75  074 

75  082 

75  089  75  097 

75  105 

75  113 

75  120 

564 

75  128 

75  136 

75  143 

75151 

75  159 

75  166  75  174 

75  182 

75  189 

75  197 

565 

75  205 

75  213 

75  220 

75  228 

75  236 

75  243  75  251 

75  259 

75  266 

75  274 

566 

75  282 

75  289 

75  297 

75  305 

75  312 

75  320  75  328 

75  335 

75  343 

75  351 

567 

75  358 

75  366 

75  374 

75  381 

75  389 

75  397  75  404 

75  412 

75  420 

75  427 

568 

75  435 

75  442 

75  450 

75  458 

75  465 

75  473  75  481 

75  4S8 

75  496 

75  504 

569 

75  511 

75  519 

75  526 

75  534 

75  542 

75  549~  75  557 

75  565 

75  572 

75  5SO 

570 

75  5S7 

75  595 

75  603 

75  610 

75  618 

75  626  75  633 

75  641 

75  648 

75  656 

571 

75  664 

75  671 

75  679 

75  686 

75  694 

75  702  75  709 

75  717 

75  724 

75  732 

572 

75  740 

75  747 

75  755 

75  762 

75  770 

75  778  75  785 

75  793 

75SOO 

75  SOS 

573 

75S15 

75  823 

75  831 

75  838 

75  846 

75  853  75  861 

75  868 

75  876 

75  SS4 

574 

75  891 

75  899 

75  906 

75  914 

75  921 

75  929  75  937 

75  944 

75  952 

75  959 

575 

75  967 

75  974 

75  982 

75  989 

75  997 

76  005  76  012 

76  020 

76027 

76  035 

576 

76  042 

76  050 

76057 

76  065 

76  072  * 

76  080  76  087 

76  095 

76103 

76110 

577 

76118 

76125 

76133 

76140 

76148 

76155  76163 

76170 

76178 

761S5 

578 

76193 

76  200 

76  208 

76  215 

76  223 

76  230  76  23S 

76  245 

76  253 

76  260 

579 

76  268 

76  275 

76  283 

76  290 

76  298 

76  305  76  313 

76  320 

76  328 

76  335 

580 

76  343 

76  350 

76  358 

76  365 

76  373 

76  380  76  388 

76  395 

76  403 

76  410 

581 

76  418 

76  425 

76  433 

76  440 

76  448 

76  455  76  462 

76470 

76  477 

76  485 

582 

76  492 

76  500 

76  507 

76  515 

76  522 

76  530  76  537 

76  545 

76  552 

76  559 

583 

76  567 

76  574 

76  582 

76  589 

76  597 

76  604  76  612 

76  619 

76  626 

76  634 

584 

76641 

76  649 

76  656 

76  664 

76  671 

76  678  76  686 

76  693 

76  701 

76  708 

585 

76  716 

76  723 

76  730 

76  738 

76  745 

76  753  76  760 

76  768 

76  775 

76  782 

586 

76  790 

76  797 

76  805 

76  812 

76  819 

76  827  76  834 

76  842 

76  849 

76  856 

587 

76  864 

76  871 

76  879 

76  886 

76  893 

76  901  76  908 

76  916 

76  923 

76  930 

588 

76938 

76  945 

76953 

76960 

76  967 

76  975  76  982 

76  989 

76  997 

77  004 

589 

77012 

77  019 

77  026 

77  034 

77  041 

77  048  77  056 

77  063 

77  070 

77  078 

590 

77085 

77  093 

77100 

77107 

77115 

77122  77129 

77137 

77144 

77151 

591 

77159 

77166 

77173 

77181 

77188 

77195  77  203 

77  210 

77  217 

77  225 

592 

77  232 

77  240 

77  247 

77  254 

77  262 

77  269  77  276 

77  283 

77  291 

77  298 

593 

77  305 

77  313 

77  320 

77  327 

77  335 

77  342  77  349 

77  357 

77  364 

77  371 

594 

77  379 

77  386 

77  393 

77  401 

77  408 

77  415  77422 

77430 

77  437 

77  444 

595 

77  452 

77  459 

77  466 

77  474 

77  481 

77  488  77495 

77  503 

77  510 

77  517 

596 

77  52£ 

77  532 

77  539 

77  546 

77  554 

77  561  77  568 

77  576 

77  583 

77  590 

597 

77  597 

77  605 

77  612 

77  619 

77  627 

77  634  77  641 

77  648 

77  656 

77  663 

598 

77  670 

77  677 

77  685 

77  692 

77  699 

77  706  77  714 

77  721 

77  728 

77  735 

599 

77  743 

77  7£0 

77  757 

77  764 

77  772 

77  779  77  786 

77  793 

77  801 

77  808 

600 

77  815 

77  822 

77  830 

77  837 

77  844 

77  851  77  859 

77S66 

77  873 

77  880 

N 

0 

1 

2 

3 

4 

5     6 

7 

8 

9 

550-600 


12 

600-650 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

600 

77  815 

77  822 

77  830 

77  837 

77  844 

77  851 

77  859 

77  866 

77  873 

77  880 

601 

77  8S7 

77  895 

77  902 

77  909 

77  916 

77  924 

77  931 

77  938 

77  945 

77  952 

602 

77  960 

77  967 

77  974 

77  981 

77  988 

77  996 

78  003 

78  010 

78  017 

78  025 

603 

78  032 

78  039 

78  046 

78  053 

78  061 

78  068 

78  075 

78  082 

78  089 

78  097 

604 

78104 

78111 

78118 

78125 

78132 

78140 

78147 

78154 

78161 

78168 

605 

78176 

73183 

78190 

78197 

78  204 

78  211 

78  219 

78  226 

78  233 

78  240 

606 

78  247 

78  254 

78  262 

78  269 

78  276 

78  283 

78  290 

78  297 

78  305 

78  312 

607 

78319 

78  326 

78  333 

78  340 

78  347 

78  355 

78  362 

78  369 

78  376 

78  383 

608 

78  390 

78  398 

78  405 

78  412 

78  419 

78  426 

78  433 

78  440 

78447 

78  455 

609 

78  462 

78469 

78  476 

78  483 

78  490 

78  497 

78  504 

78  512 

78  519 

78  526 

610 

78  533 

78  540 

78  547 

78  554 

78  561 

78  569 

78  576 

78  583 

78  590 

78  597 

611 

78  604 

78  611 

78  618 

78  625 

78  633 

78  640 

78  647 

78  654 

78  661 

78  668 

612 

78  675 

78  682 

78  689 

78  696 

78  704 

78  711 

78  718 

78  725 

78  732 

78  739 

613 

78  746 

78  753 

78  760 

78  767 

78  774 

78  781 

78  789 

78  796 

78  803 

7S810 

614 

78  817 

78  824 

78  831 

78  838 

78  845. 

78  852 

78  859 

78  866 

78  873 

78  880 

615 

78  8S8 

78  895 

78  902 

78  909 

78  916 

78  923 

78  930 

78  937 

78  944 

78  951 

616 

78  958 

78  965 

78  972 

78  979 

78  986 

78  993 

79  000 

79  007 

79  014 

79  021 

617 

79  029 

79  036 

79  043 

79  050 

79  057 

79  064 

79  071 

79  078 

79  085 

79092 

618 

79  099 

79106 

79113 

79120 

79127 

79134 

79141 

79148 

79155 

79162 

619 

79169 

79176 

79183 

79190 

79197 

79  204 

79  211 

79  218 

79  225 

79  232 

620 

79  239 

79  246 

79  253 

79  260 

79  267 

79  274 

79  281 

79  288 

79  295 

79  302 

621 

79  309 

79  316 

79  323 

79  330 

79  337 

79  344 

79  351 

79  358 

79  365 

79  372 

622 

79  379 

79  386 

79  393 

79  400 

79  407 

79  414 

79  421 

79  428 

79  435 

79  442 

623 

79  449 

79  456 

79  463 

79  470 

79  477 

79  484 

79  491 

79  498 

79  505 

79  511 

624 

79  518 

79  525 

79  532 

79  539 

79  546 

79  553 

79  560 

79  567 

79  574 

79  581 

625 

79  588 

79  595 

79  602 

79  609 

79  616 

79  623 

79  630 

79  637 

79  614 

79  650 

626 

79  657 

79  664 

79  671 

79  678 

79  685 

79  692 

79  699 

79  706 

79  713 

79  720 

627 

79  727 

79  734 

79  741 

79  748 

79  754 

79  761 

79  768 

79  775 

79  782 

79  7S9 

628 

79  796 

79  803 

79  810 

79  817 

79  824 

79  831 

79  837 

79  844 

79  851 

79  858 

629 

79  865 

79  872 

79  879 

79  886 

79  893 

79  900 

79  906 

79  913 

79  920 

79927 

630 

79  934 

79  941 

79  948 

79955 

79  962 

79  969 

79  975 

79982 

79  989 

79  996 

631 

80  003 

80  010 

80  017 

80  024 

80  030 

80  037 

80  044 

80  051 

80  058 

80  065 

632 

80  072 

80  079 

80  085 

80  092 

80  099 

80106 

80113 

80120 

SO  127 

80134 

633 

80140 

80147 

80154 

80161 

80168 

80175 

80182 

80188 

80195 

80  202 

634 

80  209 

80  216 

80  223 

80  229 

80  236 

80  243 

80  250 

80  257 

80  264 

80  271 

635 

80  277 

80  284 

80  291 

80  298 

80  305 

80  312 

80  318 

80  325 

80  332 

80  339 

636 

80  346 

80  353 

80359 

80  366 

80  373 

80  380 

80  387 

80  393 

80  400 

80  407 

637 

80414 

80  421 

80428 

80434 

80  441 

80  448 

80  455 

80  462 

80  468 

80  475 

638 

80  482 

80489 

80496 

80  502 

80  509 

80  516 

80  523 

80  530 

80  536 

80  543 

639 

80  550 

80  557 

80  564 

80  570 

80  577 

80  584 

80  591 

80  59S 

80  604 

SO  611 

640 

80  618 

80  625 

80  632 

80  638 

80  645 

80  652 

80  659 

80  665 

80  672 

SO  679 

641 

80686 

80  693 

80  699 

80  706 

80  713 

80  720 

80  726 

80  733 

80  740 

80  747 

642 

80  754 

80  760 

80  767 

80  774 

80  781 

80  787 

80  794 

80  SOI 

80  SOS 

S0S14 

643 

80  821 

80  828 

80  835 

80  841 

80  848 

80  855 

80  862 

S0S6S 

80S75 

80SS2 

644 

80  889 

80  895 

80902 

80909 

80  916 

80922 

80929 

80  936 

80  943 

SO  949 

645 

80956 

80963 

80969 

80  976 

80983 

80990 

80  996 

81003 

81010 

81017 

646 

81023 

81030 

81037 

81043 

81050 

81057 

81064 

SI  070 

SI  077 

S10S4 

647 

81090 

81097 

81104 

81111 

81117 

81124 

81131 

81137 

81144 

81151 

648 

81158 

81164 

81171 

81178 

81184 

81191 

S119S 

SI  204 

81211 

SI  218 

649 

81224 

81231 

81238 

81245 

81251 

SI  258  81265 

81271 

81278 

812S5 

650 

81291 

81298 

81305 

81311 

SI  318 

SI  325 

81331 

81338  81345 

SI  351 

H 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

600  -  650 


650  -  700 


13 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

650 

81291 

81298 

81305 

81311 

81318 

81325 

81331 

81338 

81345 

81351 

651 

81358 

81365 

81371 

81378 

81385 

81391 

81398 

81405 

81411 

81418 

652 

81425 

81431 

81438 

81445 

81451 

81458 

81465 

81471 

81478 

81485 

653 

81491 

81498  81505 

81511 

81518 

81525 

81531 

81538 

81  544 

81551 

654 

81558 

81564 

81571 

81578 

81584 

81591 

81598 

81604 

81611 

81617 

655  ■ 

SI  624 

81631 

81637 

81644 

81651 

81657 

81664 

81671 

81677 

81684 

656 

SI  690 

81697 

81704 

81710 

81717 

81723 

81730 

81737 

81743 

81750 

657 

81757 

81763 

81  770 

81776 

81783 

81790 

81796 

81803 

SI  809 

81816 

658 

81823 

81829 

81836 

81842 

81849 

81856 

81862 

81869 

81875 

81882 

659 

SI  889  81895 

81902 

81908 

81915 

81921 

81928 

81935 

81941 

81948 

660 

81954 

81961 

81968 

81974 

81981 

81987 

81994 

82  000 

82  007 

82  014 

661 

S2  020 

S2  027 

82  033 

82  040 

82  046 

82  053 

82  060 

82  066 

82  073 

82  079 

662 

82  086 

82  092 

82  099 

82  105 

82112 

82119 

82  125 

82132 

82  138 

82  145 

663 

S2  151 

82  15S 

82  164 

82  171 

82  178 

82  184 

82191 

82  197 

82  204 

82  210 

664 

S2  217 

82  223 

82  230 

82  236 

82  243 

82  249 

82  256 

82  263 

82  269 

82  276 

665 

82  282 

82  2S9 

82  295 

82  302 

82  30S 

82  315 

82  321 

82  328 

82  334 

82  341 

666 

S2  347 

82  354 

82  360 

82  367 

82  373 

82  380 

82  387 

82  393 

82  400 

82  406 

667 

82  413 

82  419 

82  426 

82  432 

82  439 

82  445 

82  452 

82  458 

82  465 

82  471 

668 

82  478 

82  484 

82  491 

82  497 

82  504 

82  510 

82  517 

82  523 

S2  530 

82  536 

669 

82  543 

82  549 

82  556 

82  562 

82  569 

82  575 

82  582 

82  588 

82  595 

82  601 

670 

82  607 

82  614 

82  620 

82  627 

82  633 

82  640 

82  646 

82  653 

82  659 

82  666 

671 

82  672 

82  679 

82  685 

82  692 

82  69S 

82  705 

82  711 

82  718 

82  724 

82  730 

672 

82  737 

82  743 

82  750 

82  756 

82  763 

82  769 

82  776 

82  782 

82  789 

82  795 

673 

82  802 

82  808 

82  814 

82  821 

82  827 

82  834 

82  840 

82  847 

82  853 

82  860 

674 

82  866 

82  872 

82  87? 

82  885 

82  892 

82  898 

82  905 

82  911 

82  918 

82  924 

675 

82  930 

82  937 

82  943 

82  950 

82  956 

82  963 

82  969 

82  975 

S2  9S2 

82  988 

676 

82  995 

83tX)l 

83  00S 

83  014 

83  020 

83  027 

83  033 

83  040 

83  046 

83  052 

677 

S3  059 

83  065 

83  072 

83  078 

83  085 

83  091 

83  097 

83  104 

83  110 

83  117 

678 

83  123 

83  129 

83  136 

83  142 

83  149 

83155 

83  161 

83  168 

83  174 

83  181 

679 

S3  187 

83  193 

83  200 

83  206 

83  213 

83  219 

83  225 

83  232 

83  23S 

83  245 

680 

83  251 

83  257 

83  264 

83  270 

83  276 

83  283 

83  289 

83  296 

83  302 

83  30S 

681 

83  315 

83  321 

83  327 

83  334 

83  340 

83  347 

83  353 

83  359 

83  366 

83  372 

682 

83  378 

83  38i 

83  391 

83  398 

83  404 

83  410 

83  417 

83  423 

S3  429 

83  436 

683 

83  442 

83  448 

83  455 

83  461 

83  467 

83  474 

83  480 

83  487 

83  493 

83  499 

684 

S3  506 

83  512 

83  518 

83  525 

83  531 

83  537 

83  544 

83  550 

83  556 

83  563 

685 

S3  569 

83  575 

83  582 

83  588 

83  594 

83  601 

83  607 

83  613 

83  620 

83  626 

686 

S3  632 

83  639 

83  645 

83  651 

83  658 

83  664 

83  670 

83  677 

83  683 

83  689 

687 

83  696 

83  702 

83  708 

83  715 

83  721 

83  727 

83  734 

83  740 

83  746 

83  753 

688 

83  759 

83  765 

83  771 

83  778 

83  784 

83  790 

83  797 

83  803 

83  809 

83  816 

689 

83  822 

83  828 

83  835 

83  841 

83  847 

83  853 

83  860 

83  866 

83  872 

83  879 

690 

83  885 

83  891 

83  897 

83  904 

83  910 

83  916 

83  923 

83  929 

83  935 

83  942 

691 

83  948 

83  954 

83  960 

83  967 

83  973 

83  979 

83  985 

83  992 

83  998 

84  004 

692 

84  011 

84  017 

84  023 

84  029 

84  036 

84  042 

84  048 

84  055 

84  061 

84  067 

693 

84  073 

84  080 

84  0S6 

84  092 

84  098 

84  105 

84111 

84117 

84123 

84130 

694 

84136 

84142 

8414S 

84  155 

S4161 

84167 

84173 

84180 

84  186 

84192 

695 

84198 

84  205 

84  211 

84  217 

84  223 

84  230 

84  236 

84  242 

84  24S 

84  255 

696 

84  261 

84  267 

84  273 

84  280 

84  286 

84  292 

84  298 

84  305 

84  311 

84  317 

697 

84  323 

84  330 

84  336 

84  342 

84  348 

84  354 

84  361 

84  367 

84  373 

84  379 

698 

84  386 

84  392 

84  398 

84  404 

84  410 

84  417 

84  423 

84  429 

84435 

84  442 

699 

84  448 

84  454 

84  460 

84  466 

84  473 

84  479 

84  485 

84  491 

84  497 

84  504 

700 

84  510 

84  516 

84  522 

84  528 

84  535 

84  541 

84  547 

84  553 

84  559 

84  566 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

650  -  700 


14 


700-750 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

700 

84  510 

84  516 

84  522 

84  528 

84  535 

S4  541 

S4  547 

84  553 

84  559 

84  566 

701 

S4  572 

84  578 

84  584 

84  590 

84  597 

84  603 

S4  609 

S4  615 

84  621 

84  628 

702 

84  634 

84  640 

84  646 

84  652 

84  658 

84  665 

84  671 

84  677 

84  683 

84  689 

703 

84  696 

84  702 

84  708 

84  714 

84  720 

84  726 

84  733 

84  739 

84  745 

S4  751 

704 

84  757 

84  763 

84  770 

84  776 

84  782 

84  788 

84  794 

84  800 

84  807 

84  813 

705 

84  819 

84  825 

84  831 

84  837 

84  844 

84  850 

84  856 

84  862 

84  868 

84  874 

706 

84  8S0 

84  887 

84  893 

84  899 

84  905 

84  911 

84  917 

84  924 

84  930 

S4  936 

767 

84  942 

84  948 

84  954 

84  960 

84  967 

84  973 

84  979 

84  985 

84  991 

84  997 

708 

85  003 

85  009 

85  016 

85  022 

85  028 

85  034 

85  040 

85  046 

85  052 

85  05S 

709 

85  065 

85  071 

85  077 

85  083 

85  089 

85  095 

85  101 

85  107 

85  114 

S5  120 

710 

85126 

85132 

85  138 

85  144 

85  150 

85156 

85  163 

S5  169 

85  175 

85  181 

711 

85  187 

85  193 

85  199 

85  205 

85  211 

85  217 

85  224 

85  230 

85  236 

S5  242 

712 

85  248 

85  254 

85  260 

85  266 

85  272 

85  278 

85  285 

85  291 

85  297 

85  303 

713 

85  309 

85  315 

85  321 

85  327 

85  333 

85  339 

85  345 

85  352 

85  35S 

85  364 

714 

85  370 

85  376 

85  382 

85  388 

85  394 

85  400 

85  406 

85  412 

85  418 

85  425 

715 

85  431 

85  437 

S5  443 

85  449 

85  455 

85  461 

85  467 

85  473 

85  479 

85  485 

716 

85  491 

85  497 

85  503 

85  509 

85  516 

85  522 

85  528 

85  534 

85  540 

85  546 

717 

85  552 

85  558 

85  564 

85  570 

85  576 

85  582 

85  5S8 

S5  594 

85  600 

85  606 

718 

85  612 

85  618 

85  625 

85  631 

85  637 

85  643 

85  649 

85  655 

85  661 

85  667 

719 

85  673 

85  679 

S5  685 

85  691 

85  697 

85  703 

85  709 

85  715 

85  721 

85  727 

720 

85  733 

85  739 

85  745 

85  751 

85  757 

85  763 

85  769 

85  775 

85  781 

85  788 

721 

85  794 

85S00 

85  806 

85  812 

85  818 

85  824 

85  830 

85S36 

85S42 

S5  848 

722 

85  854 

85  860 

S5  866 

85  872 

85  878 

85  884 

85  890 

85  896 

85  902 

S5  908 

723 

85  914 

85  920 

85  926 

85  932 

85  938 

85  944 

85  950 

85  956 

85  962 

S5  968 

724 

85  974 

85  980 

85  986 

85  992 

85  998 

86  004 

86  010 

86  016 

86  022 

86  028 

725 

S6  034 

86  040 

86  046 

86  052 

86  058 

86  064 

86  070 

86  076 

86  082 

S6  0S8 

726 

86  094 

86100 

86106 

S6112 

86118 

86124 

86130 

86136 

86141 

86147 

727 

86153 

86159 

86165 

86171 

86177 

86183 

86189 

S6195 

S6  201 

86  207 

728 

86  213 

86  219 

86  225 

86  231 

86  237 

86  243 

86  249 

86  255 

86  261 

86  267 

729 

86  273 

86  279 

86  285 

86  291 

86  297 

86  303 

86  30S 

86  314 

86  320 

86  326 

730 

86  332 

86  338 

86  344 

86  350 

86  356 

86  362 

86  368 

86  374 

86  380 

S6  3S6 

731 

86  392 

86  39S 

86  404 

86  410 

86415 

86  421 

86  427 

86  433 

86  439 

86  445 

732 

86451 

86457 

86  463 

S6  469 

86475 

86  481 

86  487 

S6  493 

86  499 

86  504 

733 

86  510 

86  516 

86  522 

86  528 

86  534 

86  540 

86  546 

86  552 

86  55S 

S6  564 

734 

86  570 

86  576 

86  581 

86  587 

86  593 

86  599 

86  605 

86  611 

86  617 

S6  623 

735 

86  629 

86  635 

86  641 

86  646 

86  652 

86  658 

S6  664 

86  670 

86  676 

86  682 

736 

86  688 

86  694 

86  700 

86  705 

86  711 

86  717 

86  723 

S6  729 

86  735 

86  741 

737 

86  747 

86  753 

86  759 

86  764 

86  770 

86  776 

S6  7S2 

S6  7SS 

S6  794 

86  800 

738 

86  806 

86  812 

86  817 

86  823 

86  829 

86S35 

86  841 

86S47 

86  853 

S6  859 

739 

86  864 

86  870 

86  876 

86  882 

86  888 

86  894 

86  900 

86  906 

86  911 

86  917 

740 

86  923 

S6  929 

86  935 

86  941 

86  947 

86  953 

86  958 

86  964 

86  970 

S6  976 

741 

86  9S2 

86  988 

86  994 

86  999 

87  005 

87  011 

87  017 

87  023 

87  029 

87  035 

742 

87  040 

87  046 

87  052 

87  058 

87  064 

87  070 

87  075 

87  0S1 

87  087 

S7  093 

743 

87  099 

87105 

87111 

87116 

S7122 

87128 

87134 

S7140 

S7  146 

87151 

744 

87157 

87163 

87169 

87175 

87181 

S71S6 

87192 

87198 

87  204 

S7  210 

745 

87  216 

87  221 

87  227 

S7  233 

87  239 

87  245 

87  251 

87  256 

87  262 

87  268 

746 

87  274 

87  280 

S7  286 

87  291 

87  297 

S7  303 

S7  309 

87  315 

S7  320 

S7  326 

747 

87  332 

87  338 

S7  344 

87  349 

87  355 

S7  361 

87  367 

S7  373 

87  379 

S7  384 

748 

87  390 

87  396 

S7  402 

87  408 

87  413 

87  419 

87  425 

S7  431 

87  437 

S7  442 

749 

87  448 

87  454 

S7  460 

87  466 

87  471 

87  477 

S7  483 

87  489  87  495 

S7  500 

750 

87  506 

87  512 

87  518 

87  523 

87  529 

87  535 

87  541 

87  547 

87  552 

87  55S 

0 

1 

2 

3 

4 

o 

6 

7 

8 

9 

700  -  750 


750-800 

15 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

750 

87  506 

87  512 

87  518 

87  523 

87  529 

87  535 

87  541 

87  547 

87  552 

87  558 

751 

87  564 

87  570 

87  576 

87  581 

87  587 

87  593 

87  599 

87  604 

87  610 

87  616 

752 

87  622 

87  628 

87  633 

87  639 

87  645 

87  651 

87  656 

87  662 

87  668 

87  674 

753 

87  679 

S7  685 

87691 

87  697 

87  703 

87  708 

87  714 

87  720 

87  726 

87  731 

754 

87  737 

87  743 

87  749 

87  754 

87  760 

87  766 

87  772 

87  777 

87  783 

87  789 

755 

87  795 

87  800 

87  806 

87812 

87  818 

87  823 

87  829  87  835 

87  841 

87  846 

756 

87  852 

87  858 

87  864 

87  869 

87  875 

87  881 

87  887 

87  892 

87  898 

87  904 

757 

87  910 

87  915 

S7  921 

87927 

87  933 

87  938 

87944 

87  950 

87  955 

87  961 

758 

87967 

87973 

87978 

87984 

87990 

87  996 

88  001 

88  007 

88  013 

88  018 

759 

88  024 

88030 

88  036 

88  041 

88  047 

88  053 

88  058 

88  064 

8S070 

88  076 

760 

88  081 

88  087 

88093 

88  098 

88104 

88110 

88116 

88121 

88127 

88133 

761 

8813S 

88144 

88  150 

88156 

88161 

88167 

88173 

88178 

88184 

88190 

762 

88195 

88  201 

88  207 

88  213 

88  218 

88  224 

88  230 

88  235 

88  241 

88  247 

763 

88  252 

SS25S 

88  264 

88  270 

88  275 

88  281 

88  287 

88  292 

88  29S 

88  304 

764 

88  309 

88  315 

88321 

88  326 

88  332 

88338 

88343 

88  349 

88  355 

88  360 

765 

88  366 

88372 

88  377 

88  383 

88  389 

88  395 

88  400 

88  406 

88  412 

88  417 

766 

88  423 

SS429 

88  434 

88  440 

88  446 

88451 

88457 

88  463 

88  468 

88  474 

767 

8S4S0 

SS4S5 

88491 

88  497 

88  502 

88  508 

88  513 

88  519 

88  525 

88  530 

768 

88  536 

88  542 

88  547 

88  553 

88  559 

88  564 

88  570 

88  576 

88  581 

88  587 

769 

8S593 

S8  598 

88  604 

88  610 

88  615 

88  621 

88  627 

88  632 

88  638 

88  643 

770 

88  649 

88655 

8S660 

88  666 

88  672 

88  677 

88  683 

8S6S9 

88  694 

88  700 

771 

8S705 

88  711 

S8  717 

88  722 

88  728 

88  734 

88  739 

88  745 

88  750 

S8  756 

772 

88  762 

8S767 

88  773 

88  779 

8S7S4 

88  790 

88  795 

88  801 

88  807 

88  812 

773 

88  818 

S8  824 

S8  829 

8SS35 

88  840 

88  846 

88  852 

88  857 

88  863 

8S868 

774 

S8  874 

SSSSO 

88  885 

88  891 

88  897 

8S902 

S8  908 

S8  913 

88  919 

88  925 

775 

88930 

88936 

88  941 

88947 

88  953 

88  958 

88  964 

88  969 

88  975 

88  981 

776 

88  986 

88  992 

88  997 

89  003 

89  009 

89  014 

89  020 

89  025 

89  031 

89  037 

777 

S9  042 

89  04S 

89  053 

89  059 

89  064 

89  070 

89  076 

89  081 

89  087 

89  092 

778 

89  098 

89104 

S9109 

89115 

89120 

89126 

89131 

89137 

89143 

8914S 

779 

89154 

89159 

89165 

89170 

89176 

89182 

89187 

89193 

89198 

89  204 

780 

89  209 

89  215 

89  221 

89  226 

89232 

89  237 

89  243 

89  248 

89  254 

89  260 

781 

89  265 

89  271 

89  276 

89  282 

89  287 

89  293 

89  298 

89  304 

89  310 

89  315 

782 

89  321 

89  326 

89  332 

89  337 

89  343 

89  348 

89  354 

89  360 

89  365 

89  371 

783 

89  376 

89  382 

89  387 

89  393 

89  398 

89  404 

89  409 

89  415 

89  421 

89  426 

784 

89432 

89437 

89443 

89448 

89  454 

89459  89  465 

89  470 

89  476 

89  481 

785 

89  487 

89492 

89  498 

89  504 

89  509 

89  515 

89  520 

89  526 

89  531 

89  537 

786 

89  542 

89  548 

89  553 

89  559 

89  564 

89  570 

89  575 

89  581 

89  586 

89  592 

787 

89  597 

89  603 

89  609 

89  614 

89  620 

89  625 

89  631 

89  636 

89  642 

89  647 

788 

89  653 

89  658 

89  664 

89  669 

89  675 

89  680 

89  686 

89  691 

89  697 

89  702 

789 

89  708 

89  713 

89  719 

89  724 

89  730 

89  735 

89  741 

89  746 

89  752 

89  757 

790 

89  763 

89  768 

89  774 

89  779 

89  785 

89  790 

89  796 

89  801 

89  807 

89  812 

791 

89  818 

89  823 

89  829 

89  834 

89  840 

89  845 

89  851 

89  856 

89  862 

89  867 

792 

89  873 

89  878 

89  883 

89  889 

89  894 

89  900 

89  905 

89  911 

89  916 

89  922 

793 

89  927 

89933 

89  938 

89  944 

89  949 

89  955 

89  960 

89  966 

89  971 

89  977 

794 

89  982 

89988 

89  993 

89  998 

90004 

90  009 

90  015 

90  020 

90  026 

90  031 

795 

90  037 

90042 

90  048 

90  053 

90  059 

90  064 

90  069 

90  075 

90  080 

90086 

796 

90  091 

90  097 

90102 

90108 

90113 

90119 

90124 

90  129 

90135 

90140 

797 

90146 

90151 

90157 

90162 

90168 

90173 

90179 

90184 

90189 

90195 

798 

90  200 

90  206 

90  211 

90  217 

90  222 

90  227 

90  233 

90  238 

90  244 

90  249 

799 

90  255 

90  260 

90  266 

90  271 

90  276 

90  282 

90  287 

90  293 

90  298 

90  304 

800 

90  309 

90  314 

90  320 

90  325 

90  331 

90  336 

90  342 

90  347 

90  352 

90  35S 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

750  -  800 


16 

800-850 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

800 

90  309 

90  314 

90  320 

90  325 

90  331 

90336 

90  342 

90  347 

90  352 

90  358 

801 

90  363 

90  369  90  374 

90  380 

90  385 

90  390 

90  396 

90  401 

90  407 

90412 

802 

90417 

90  423 

90  428 

90  434 

90439 

90  445 

90  450 

90  455 

90  461 

90  466 

803 

90  472 

90  477 

90  482 

90  488 

90  493 

90  499 

90  504 

90  509 

90  515 

90  520 

804 

90  526 

90  531 

90  536 

80  542 

90  547 

90  553 

90  558 

90  563 

90  569 

90  574 

805 

90  580 

90  585 

90  590 

90  596 

90  601 

90  607 

90  612 

90  617 

90  623 

90628 

806 

90  634 

90  639 

90  644 

90  650 

90  655 

90  660 

90  666 

90  671 

90  677 

90  682 

807 

90  687 

90  693 

90  698 

90  703 

90  709 

90  714 

90  720 

90  725 

90  730 

90  736 

808 

90  741 

90  747 

90  752 

90  757 

90  763 

90  768 

90  773 

90  779 

90  784 

90  789 

809 

90  795 

90  800 

90  806 

90  811 

90  816 

90  822 

90  827 

90  832 

90  838 

90  843 

810 

90  849 

90  854 

90  859 

90  865 

90  870 

90  875 

90  8S1 

90  886 

90  891 

90  897 

811 

90  902 

90  907 

90  913 

90  918 

90  924 

90929 

90  934 

90  940 

90  945 

90950 

812 

90  956 

90  961 

90  966 

90  972 

90  977 

90  982 

90  988 

90  993 

90  998 

91004 

813 

91009 

91014 

91020 

91025 

91030 

91036 

91041 

91046 

91052 

91057 

814 

91062 

9106S 

91073 

91078 

91084 

91089 

91094 

91100 

91105 

91110 

815 

91116 

91121 

91126 

91132 

91137 

91142 

91148 

91153 

91 158 

91164 

816 

91169 

91174 

91180 

91185 

91190 

91196 

91201 

91206 

91212 

91217 

817 

91222 

91228 

91233 

91238 

91243 

91249 

91254 

91259 

91265 

91270 

818 

91275 

91281 

91286 

91291 

91297 

91302 

91307 

91312 

91318 

91323 

819 

91328 

91334 

91339 

91344  91350 

91355 

91360 

91365 

91371 

91376 

820 

91381 

91387 

91392 

91397 

91403 

91408 

91413 

91418 

91424 

91429 

821 

91434 

91440 

91445 

91450 

91455 

91461 

91466 

91471 

91477 

91482 

822 

91487 

91492 

91498 

91503 

91508 

91514 

91519 

91524 

91529 

91535 

823 

91540 

91545 

91551 

91556 

91561 

91566 

91572 

91577 

91582 

91587 

824 

91593 

91598 

91603 

91609 

91614 

91619 

91624 

91630  91635 

91640 

825 

91645 

91  651 

91656 

91661 

91666 

91672 

91677 

91682 

91687 

91693 

826 

91698 

91703 

91709 

91714 

91719 

91724 

91730  91735 

91740 

91745 

827 

91751 

91756 

91761 

91766 

91772 

91777 

91782 

91787 

91793 

91798 

828 

91803 

91808 

91814 

91819 

91824 

91829 

91834 

91840 

91845 

91850 

829 

91855 

91861 

91866 

91871 

91876 

91882 

91887 

91892 

91897 

91903 

830 

91908 

91913 

91918 

91924 

91929 

91934 

91939 

91944 

91950 

91955 

831 

91960 

91965 

91971 

91976 

91981 

91986 

91991 

91997 

92  002 

92  007 

832 

92  012 

92  018 

92  023 

92  028 

92  033 

92  038 

92  044 

92  049 

92  054 

92  059 

833 

92  065 

92  070 

92  075  92  080  92  085 

92  091 

92  096 

92  101 

92  106 

92111 

834 

92117 

92122 

92127 

92132 

92137 

92143 

92148 

92  153 

92158 

92  163 

835 

92169 

92174 

92179 

92184 

92189 

92195 

92  200 

92  205 

92  210 

92  215 

836 

92  221 

92  226 

92  231 

92  236 

92  241 

92  247 

92  252 

92  257 

92  262 

92  267 

837 

92  273 

92  278 

92  283 

92  288 

92  293 

92  298 

92  304 

92  309 

92  314 

92  319 

838 

92  324 

92  330 

92  335 

92  340 

92  345 

92  350 

92  355 

92  361 

92  366 

92  371 

839 

92  376 

92  381 

92  387 

92  392 

92  397 

92  402 

92  407 

92  412 

92  418 

92  423 

840 

92  428 

92  433 

92  438 

92  443 

92  449 

92  454 

92  459 

92  464 

92  469 

92  474 

841 

92  480  92  485 

92  490 

92  495 

92  500 

92  505 

92  511 

92  516 

92  521 

92  526 

842 

92  531 

92  536 

92  542 

92  547 

92  552 

92  557 

92  562 

92  567 

92  572 

92  578 

843 

92  583 

92  5S8 

92  593 

92  598 

92  603 

92  609 

92  614 

92  619 

92  624 

92  629 

844 

92  634 

92  639 

92  645 

92  650 

92  655 

92  660 

92  665 

92  670 

92  675 

92  681 

845 

92  686 

92  691 

92  696 

92  701 

92  706 

92  711 

92  716 

92  722 

92  727 

92  732 

846 

92  737 

92  742 

92  747 

92  752 

92  758 

92  763 

92  768 

92  773 

92  77S 

92  7S3 

847 

92  788 

92  793 

92  799 

92  804 

92  809 

92  814 

92  819 

92  824 

92S29 

92S34 

848 

92  840  92  845. 

92  850  92  855 

92  860 

92  865 

92  870 

92S75 

92  SSI 

92  886 

849 

92  891 

92  896 

92  901 

92  906 

92  911 

92  916 

92  921 

92  927 

92  932 

92  937 

850 

92  942 

92  947 

92  952 

92  957 

92  962 

92  967 

92  973 

92  978 

92  983 

92  98S 

[■ 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

800-850 


850  -  1 

WO 

17 

N 

0 

1 

2 

3 

4 

5 

6 

7     8    9 

850 

92  942 

92  947 

92  952 

92  957 

92  962 

92  967 

92  973 

92  978  92  983  92  988 

851 

92  993 

92  998 

93  003 

93  008 

93  013 

93  018 

93  024 

93  029  93  034  93  039 

852 

93  044 

93  049 

93  054 

93  059 

93  064 

93  069 

93  075 

93  080  93  0S5  93  090 

853 

93  095 

93  100 

93  105 

93  110 

93115 

93120 

93125 

93  131  93  136  93141 

854 

93146 

93  151 

93  156 

93  161 

93  166 

93171 

93176 

93  181  93  186  93192 

855 

93  197 

93  202 

93  207 

93  212 

93  217 

93  222 

93  227 

93  232  93  237  93  242 

856 

93  247 

93  252 

93  258 

93  263 

93  268 

93  273 

93  278 

93  283  93  288  93  293 

857 

93  298 

93  303 

93  308 

93  313 

93  318 

93  323 

93  328 

93  334  93  339  93  344 

858 

93  349 

93  354 

93  359 

93  364 

93  369 

93  374 

93  379 

93  384  93  389  93  394 

859 

93  399 

93  404 

93  409 

93  414 

93  420 

93  425 

93  430 

93  435  93  440  93  445 

860 

93  450  93  455 

93  460 

93  465 

93  470 

93  475 

93  480 

93  485  93  490  93  495 

861 

93  500 

93  505 

93  510 

93  515 

93  520 

93  526 

93  531 

93  536  93  541  93  546 

862 

93  551 

93  556 

93  561 

93  566 

93  571 

93  576 

93  581 

93  586  93  591  93  596 

863 

93  601 

93  606 

93  611 

93  616 

93  621 

93  626 

93  631 

93  636  93  641  93  646 

864 

93  651 

93  656 

93  661 

93  666 

93  671 

93  676 

93  682 

93  687  93  692  93  697 

865 

93  702 

93  707 

93  712 

93  717 

93  722 

93  727 

93  732 

93  737  93  742  93  747 

866 

93  752 

93  757 

93  762 

93  767 

93  772 

93  777 

93  782 

93  7S7  93  792  93  797 

867 

93S02 

93  807 

93  812 

93  817 

93  822 

93  827 

93  832 

93  837  93  842  93  847 

868 

93  852 

93  857 

93  862 

93  867 

93  872 

93  877 

93  882 

93  887  93  892  93  897 

869 

93  902 

93  907 

93  912 

93  917 

93  922 

93  927 

93  932 

93  937  93  942  93  947 

870 

93  952 

93  957 

93  962 

93  967 

93  972 

93  977 

93  982 

93  987  93  992  93  997 

871 

94  002 

94  007 

94  012 

94  017 

94  022 

94  027 

94  032 

94  037  94  042  94  047 

872 

94  052 

94  057 

94  062 

94  067 

94  072 

94  077 

94  082 

94  086  94  091  94  096 

873 

94101 

94106 

94111 

94116 

94121 

94126 

94131 

94  136  94141  94146 

874 

94151 

94156 

94161 

94166 

94171 

94176 

94181 

94186  94191  94196 

875 

94  201 

94  206 

94  211 

94  216 

94  221 

94  226 

94  231 

94  236  94  240  94  245 

876 

94  250 

94  255 

94  260 

94  265 

94  270 

94  275 

94  280 

94  285  94  290  94  295 

877 

94  300  94  305 

94  310  94  315 

94  320 

94  325 

94  330  94  335  94  340  94  345 

878 

94  349 

94  354 

94  359 

94  364 

94  369 

94  374 

94  379 

94  384  94  389  94  394 

879 

94  399 

94  404 

94  409 

94  414 

94  419 

94424 

94  429 

94  433  94438  94443 

880 

94448 

94  453 

94  458 

94  463 

94  468 

94  473 

94  478 

94  483  94  488  94  493 

881 

94  498 

94  503 

94  507 

94  512 

94  517 

94  522 

94  527 

94  532  94  537  94  542 

882 

94  547 

94  552 

94  557 

94  562 

94  567 

94  571 

94  576 

94  581  94  586  94  591 

883 

94  596 

94  601 

94  606 

94  611 

94  616 

94  621 

94  626 

94  630  94  635  94  640 

884 

94  645 

94  650 

94  655 

94  660  94  665 

94  670  94  675 

94  680  94  685  94  689 

885 

94  694 

94  699 

94  704 

94  709 

94  714 

94  719 

94  724 

94  729  94  734  94  738 

886 

94  743 

94  748 

94  753 

94  758 

94  763 

94  768 

94  773 

94  778  94  783  94  787 

887 

94  792 

94  797 

94  802 

94  807 

94  812 

94  817 

94  822 

94  827  94  832  94  836 

888 

94  841 

94  846 

94  851 

94  856 

94  861 

94  866 

94  871 

94876  94  880  94  885 

889 

94  890 

94  895 

94  900  94  905 

94  910 

94  915 

94  919 

94  924  94  929  94  934 

890 

94  939 

94  944 

94  949 

94  954 

94959 

94  963 

94  968 

94  973  94  978  94  983 

831 

94  9S8 

94  993 

94  998 

95  002 

95  007 

95  012 

95  017 

95  022  95  027  95  032 

892 

95  036 

95  041 

95  046 

95  051 

95  056 

95  061 

95  066 

95  071  95  075  95  080 

893 

95  085 

95  090 

95  095  95100  95105 

95109 

95114 

95119  95124  95129 

894 

95134 

95139 

95143 

95148 

95153 

95  158 

95163 

95168  95173  95177 

895 

95  182 

95187 

95192 

95  197 

95  202 

95  207 

95  211 

95  216  95  221  95  226 

836 

95  231 

95  236 

95  240 

95  245 

95  250 

95  255 

95  260 

95  265  95  270  95  274 

897 

95  279 

95  284 

95  289 

95  294 

95  299 

95  303 

95  308 

95  313  95  318  95  323 

898 

95  328 

95  332 

95  337 

95  342 

95  347 

95  352 

95  357 

95  361  95  366  95  371 

899 

95  376 

95  381 

95  3S6 

95  390 

95  395 

95  400  95  405 

95  410  95  415  95  419 

000 

95  424 

95  429 

95  434 

95  439 

95  444 

95  448 

95  453 

95  458  95  463  95  468 

X 

0 

1 

o 

3 

4 

5 

6 

7    8    9 

850  -  900 


18 


900-950 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

900 

95  424 

95  429 

95  434 

95  439 

95  444 

95  448 

95  453 

95  458 

95  463 

95  468 

901 

95  472 

95  477 

95  482 

95  487 

95  492 

95  497 

95  501 

95  506 

95  511 

95  516 

902 

95  521 

95  525 

95  530 

95  535 

95  540 

95  545 

95  550  95  554 

95  559 

95  564 

903 

95  569 

95  574 

95  578 

95  583 

95  588 

95  593 

95  598 

95  602 

95  607 

95  612 

904 

95  617 

95  622 

95  626 

95  631 

95  636 

95  641 

95  646 

95  650 

95  655 

95  660 

905 

95  665 

95  670 

95  674 

95  679 

95  684 

95  689 

95  694 

95  698 

95  703 

95  708 

906 

95  713 

95  718 

95  722 

95  727 

95  732 

95  737 

95  742 

95  746 

95  751 

95  756 

907 

95  761 

95  766 

95  770 

95  775 

95  780 

95  785 

95  789 

95  794 

95.799 

95  804 

908 

95  809 

95  813 

95  818 

95  823 

95  828 

95  832 

95  837 

95  842 

95  847 

95  852 

909 

95  856 

95  861 

95  866 

95  871 

95  875 

95  880 

95  885 

95  890 

95  895 

95  899 

910 

95  904 

95  909 

95  914 

95  918 

95  923 

95  928 

95  933 

95  938 

95  942 

95  947 

911 

95  952 

95  957 

95  961 

95  966 

95  971 

95  976 

95  980 

95  985 

95  990 

95  995 

912 

95  999 

96  004 

96  009 

96  014 

96  019 

96  023 

96  028 

96  033 

96  038 

96  042 

913 

96047 

96  052 

96  057 

96  061 

96  066 

96  071 

96  076 

96  080 

96  085 

96090 

914 

96  095 

96  099 

96104 

96109 

96114 

96118 

96123 

96128 

96133 

96137 

915 

96142 

96147 

96152 

96156 

96161 

96166 

96171 

96175 

96180 

96185 

916 

96190 

96194 

96199 

96  204 

96  209 

96  213 

96  218 

96  223 

96  227 

96  232 

917 

96  237 

96  242 

96  246 

96  251 

96  256 

96  261 

96  265 

96  270 

96  275 

96  280 

918 

96  284 

96  289 

96  294 

96  298 

96  303 

96308 

96  313 

96  317 

96  322 

96  327 

919 

96  332 

96  336 

96  341 

96  346 

96  350 

96  355 

96  360 

96  365 

96  369 

96  374 

920 

96  379 

96  384 

96  388 

96  393 

96  398 

96402 

96407 

96  412 

96417 

96  421 

921 

96  426 

96  431 

96  435 

96  440 

96  445 

96450 

96  454 

96459 

96  464 

96  468 

922 

96  473 

96  478 

96  483 

96  487 

96  492 

96  497 

96  501 

96  506 

96  511 

96  515 

923 

96  520 

96  525 

96  530 

96  534 

96  539 

96  544 

96  548 

96  553 

96  558 

96  562 

924 

96  567 

96  572 

96  577 

96  581 

96  586 

96  591 

96  595 

96  600 

96  605 

96  609 

925 

96  614 

96  619 

96  624 

96  628 

96  633 

96  638 

96  642 

96  647 

96  652 

96  656 

926 

96  661 

96  666 

96  670 

96  675 

96  680 

96  685 

96  689 

96  694 

96  699 

96  703 

927 

96  708 

96  713 

96  717 

96  722 

96  727 

96  731 

96  736 

96  741 

96  745 

96  750 

928 

96  755 

96  759 

96  764 

96  769 

96  774 

96  778 

96  783 

96  788 

96  792 

96  797 

929 

96  802 

96  806 

96  811 

96  816 

96  820 

96  825 

96  830 

96  834 

96  839 

96  844 

930 

96  848 

96  853 

96  858 

96  862 

96  867 

96  872 

96  876 

96  881 

96  886 

96  890 

931 

96  895 

96  900 

96  904 

96  909 

96  914 

96  918 

96  923 

96  928 

96  932 

96  937 

932 

96  942 

96  946 

96  951 

96  956 

96  960 

96  965 

96  970 

96  974 

96  979 

96  984 

933 

96  988 

96  993 

96997 

97  002 

97  007 

97  011 

97  016 

97  021 

97  025 

97  030 

934 

97  035 

97  039 

97  044 

97  049 

97  053 

97  058 

97  063 

97  067 

97  072 

97  077 

935 

97  081 

97  086 

97  090 

97  095 

97100 

97104 

97109 

97114 

97118 

97123 

936 

97128 

97132 

97137 

97142 

97146 

97151 

97155 

97160 

97165 

97169 

937 

97174 

97179 

97183 

97188 

97192 

97197 

97  202 

97  206 

97  211 

97  216 

938 

97  220 

97  225 

97  230 

97  234 

97  239 

97  243 

97  248 

97  253 

97  257 

97  262 

939 

97  267 

97  271 

97  276 

97  280 

97  285 

97  290 

97  294 

97  299 

97  304 

97  308 

940 

97  313 

97  317 

97  322 

97  327 

97  331 

97  336 

97  340 

97  345 

97  350  97  354 

941 

97  359 

97  364 

97  368 

97  373 

97  377 

97  382 

97  387 

97  391 

97  396 

97  400 

942 

97  405 

97  410 

97414 

97  419 

97  424 

97  428 

97  433 

97  437 

97  442 

97  447 

943 

97  451 

97  456 

97  460 

97  465 

97  470 

97  474 

97  479 

97  483 

97  488 

97  493 

944 

97497 

97  502 

97  506 

97  511 

97  516 

97  520  97  525 

97  529 

97  534 

97  539 

945 

97  543 

97  548 

97  552 

97  557 

97  562 

97  566 

97  571 

97  575 

97  580  97  585 

946 

97  589 

97  594 

97  598 

97  603 

97  607 

97  612 

97  617 

97  621 

97  626 

97  630 

947 

97  635 

97  640 

97  644 

97  649 

97  653 

97  658 

97  663 

97  667 

97  672 

97  676 

948 

97  681 

97  685 

97  690 

97  695 

97  699 

97  704 

97  708 

97  713 

97  717 

97  722 

949 

97  727 

97  731 

97  736  97  740  97  745 

97  749 

97  754 

97  759 

97  763 

97  768 

950 

97  772 

97  777 

97  782 

97  786 

97  791 

97  795 

97  800 

97  804 

97  809 

97  813 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

900  -  950 


950-1000 


19 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

950 

97  772 

97  777 

97  782 

97  786 

97  791 

97  795 

97  800 

97  804 

97  809 

97  813 

951 

97  818 

97  823 

97  827 

97  832 

97  836 

97  841 

97  845 

97  850 

97  855 

97  859 

952 

97  864 

97  868 

97  873 

97  877 

97  882 

97  886 

97  891 

97  896 

97  900  97  905 

953 

97  909 

97  914 

97  918 

97  923 

97  928 

97  932 

97  937 

97  941 

97  946 

97  950 

954 

97  955 

97  959 

97  964 

97  968 

97  973 

97  978 

97  982 

97  987 

97  991 

97  996 

955 

98  000 

98  005 

98  009 

98  014 

98  019 

98  023 

98  028 

98  032 

98  037 

98  041 

956 

98  046  98  050  98  055 

98  059 

98  064 

98  068 

98  073 

98  078 

98  082 

98  087 

957 

98  091 

98  096  98100  98105 

98109 

98114 

98118 

98123 

98127 

98132 

958 

98137 

98141 

98146  98150  98155 

98159 

98164 

98168 

98173 

98177 

959 

98182 

98186 

98191 

98195 

98  200 

98  204 

98  209 

98  214 

98  218 

98  223 

960 

98  227 

98  232 

98  236 

98  241 

98  245 

98  250  98  254  98  259  98  263 

98  268 

961 

98  272 

98  277 

98  281 

98  286 

98  290 

98  295 

98  299 

98  304 

98  308 

98  313 

962 

98  318 

98  322 

98  327 

98  331 

98  336 

98  340 

98  345 

98  349 

98  354 

98  358 

963 

98  363 

98  367 

98  372 

98  376 

98  381 

98  385 

98  390 

98  394 

98  399 

98  403 

964 

98  408 

98  412 

98  417 

98421 

98  426 

98  430 

98  435 

98  439 

98  444 

98  448 

965 

98  453 

98  457 

98  462 

98  466 

98  471 

98  475 

98  480 

98  484 

98  489 

98  493 

966 

98  498 

98  502 

98  507 

98  511 

98  516 

98  520 

98  525 

98  529 

98  534 

98  538 

967 

98  543 

98  547 

98  552 

98  556 

98  561 

98  565 

98  570 

98  574 

98  579 

98  583 

968 

98  588 

98  592 

98  597 

98  601 

98  605 

98  610 

98  614 

98  619 

98  623 

98  628 

969 

98  632 

98  637 

98  641 

98  646 

98  650 

98  655 

98  659 

9S664 

98  668 

98  673 

970 

98  677 

98  682 

98  686 

98  691 

98  695 

98  700 

98  704 

98  709 

98  713 

98  717 

971 

98  722 

98  726 

98  731 

98  735 

98  740 

98  744 

98  749 

98  753 

98  758 

98  762 

972 

98  767 

98  771 

98  776 

98  780 

98  784 

98  789 

98  793 

98  798 

98  802 

98  807 

973 

98  811 

98S16 

98  820  98  825 

98  829 

98  834 

98  838 

98  843 

98  847 

98  851 

974 

98  856 

98S60 

98  865 

98  869 

98  874 

98  878 

98  883 

98  887 

98  892 

98  896 

975 

98  900  98  905 

98  909 

98  914 

98  918 

98  923 

98  927 

98  932 

98  936 

98  941 

976 

98  945 

98  949 

98  954 

98  958 

98  963 

98  967 

98  972 

98  976 

98  981 

98  985 

977 

98  989 

98  994 

98  998 

99  003 

99  007 

99  012 

99  016 

99  021 

99  025 

99  029 

978 

99034 

99  038 

99  043 

99  047 

99  052 

99  056 

99  061 

99  065 

99  069 

99  074 

979 

99  078 

99  083 

99  087 

99  092 

99  096 

99100 

99105 

99109 

99114 

99118 

980 

99123 

99127 

99131 

99136 

99140 

99145 

99149 

99154 

99158 

99162 

981 

99167 

99171 

99176 

99180 

99185 

99189 

99193 

99198 

99  202 

99  207 

982 

99  211 

99  216 

99  220 

99  224 

99  229 

99  233 

99  238 

99  242 

99  247 

99  251 

983 

99  255 

99  260 

99  264 

99  269 

99  273 

99  277 

99  282 

99  286 

99  291 

99  295 

984 

99  300 

99  304 

99  308 

99  313 

99  317 

99  322 

99  326  99  330  99  335 

99  339 

985 

99  344 

99  348 

99  352 

99  357 

99  361 

99  366 

99  370 

99  374 

99  379 

99  383 

986 

99  388 

99  392 

99  396 

99  401 

99  405 

99  410 

99  414 

99  419 

99  423 

99  427 

987 

99  432 

99436 

99  441 

99  445 

99  449 

99  454 

99  458 

99  463 

99  467 

99  471 

988 

99  476 

99  480 

99484 

99489 

99  493 

99  498 

99  502 

99  506 

99  511 

99  515 

989 

99  520 

99  524 

99  528 

99  533 

99  537 

99  542 

99  546 

99  550  99  555 

99  559 

990 

99  564 

99  568 

99  572 

99  577 

99  581 

99  585 

99  590 

99  594 

99  599 

99  603 

991 

99  607 

99  612 

99  616 

99  621 

99  625 

99  629 

99  634 

99  638 

99  642 

99  647 

992 

99  651 

99  656 

99  660 

99  664 

99  669 

99  673 

99  677 

99  682 

99  686 

99  691 

993 

99  695 

99  699 

99  704 

99  708 

99  712 

99  717 

99  721 

99  726 

99  730 

99  734 

994 

99  739 

99  743 

99  747 

99  752 

99  756 

99  760 

99  765 

99  769 

99  774 

99  778 

995 

99  782 

99  787 

99  791 

99  795 

99  800 

99  804 

99  808 

99  813 

99  817 

99  822 

996 

99  826 

99  830 

99  835 

99  839 

99  843 

99  848 

99  852 

99  856 

99  861 

99  865 

997 

99  870 

99  874 

99  878 

99  883 

99  887 

99  891 

99  896 

99  900 

99  904 

99  909 

998 

99913 

99  917 

99  922 

99  926 

99930 

99  935 

99  939 

99  944 

99  948 

99  952 

999 

99957 

99  961 

99  965 

99  970 

99  974 

99  978 

99  983 

99  987 

99  991 

99  996 

1000 

00  000 

00  004 

00  009 

00  013 

00  017 

00  022 

00026  00030  00  035 

00039 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

950  - 1000 


20      TABLE  II, -LOGARITHMS  OP  CONSTANTS. 


Circumference  of  the  Circle  in  degrees =  360 

Circumference  of  the  Circle  in  minutes =      21  600 

Circumference  of  the  Circle  in  seconds . =  1  296  000 

If  the  radius  r=  1,  half  the  Circumference  of  the  Circle  is 
7t  =  3. 14  159  265  358  979  323  846  264  338  328 


log 
2.  55  630  250 
4. 33  445  375 
6. 11  260  500 

0. 49  714  987 


Also: 
2t  =    6.28  318  531 
4t  =  12.  56  637  061 
1.57079  633 


2 


=    1.04  719  755 


—  =   4.18  879  020 


=  0.78  539  816 

=  0.52  359  878 

=  0.31830  989 

=  0.15  915  494 

=  0.95  492  966 

=  1.27  323  954 

=  0.23  873  241 


log 
0.79  817  987 
1.09  920  986 
0.19  611988 

0.  02  002  862 

0. 62  208  861 

9. 89  508  988  - 10 

9.  71  899  862  -  10 

9.  50  285  013  -  10 

9.  20  182  013  -  10 

9. 97  997  138  -  10 

0.10491012 

9.37  791139-10 


1 

3 

At 

4 


x 


3 

±- 

8/7T 

6 


:  9.  86  960  440 
0. 10  132  118 
1.77  245  385 
0.56  418  958 

0. 97  720  502 
1.12  837  917 
1.46  459189 
0. 68  278  406 
2. 14  502  940 
0.  62  035  049 
0.  80  599  598 


Arc  a,  whose  length  is  equal  to  the  radius  r,  is : 

in  degrees «°  ...'..  =   — =  57.29  577  951°  . 


in  minutes a' 

in  seconds a" 


re 
10  800 


7T 

648  000 


=  3  437.74  677' 
,=  206  264.806" 


Arc  2  a,  whose  length  is  equal  to  twice  the  radius, 2  r,  is 
in  degrees 2a°  . . . 


in  minutes 2ar 

in  seconds 2a" 


TV 

21  GOO 


7T 

1  296  000 


If  the  radius  r  =  1,  the  length  of  the  arc  is  : 


for  1  degree 


for  1  minute ■ 


for  1  second • 


for  £  degree....—. 

for  h  minute — -, 

2a1 


1 

2a" 


TC 

180*" 

7T 

10  800 


648  000 

TC 

360"" 


21600 


for  \  second 


114.  59  155  903 
6  875.49  354'. 
412  529. 612"  . 

=  0.01745  329.. 
=  0.00029  089.. 
=  0.00000  485.. 
-0.00  872  665... 
.=  0.00014  544.. 
=  0.00000  242.. 


1  296  000 
Sin  1"  in  the  unit  circle =  0.  00  000  485 


log 
0. 99  429  975 

9.00  570  025-10 

0.  24  857  494 

9.  75  142  506  -  10 

9.  9S  998  569  -  10 
0.  05  245  506 
0. 16  571  662 
9.83  428  338-10 
0.  33  143  325 
9.  79  263  713  -  10 
9.  90  633  287  -  10 


log 
1.  75  812  263 

3.  53  627  388 

5.31442  513 

2. 05  915  263 
3.  83  730  388 
5.61545  513 

8.  24  187  737  -  10 
6.46  372  612-10 
4.68  557  487-10 
7.94  084  737-10 
6. 16  269  612  -  10 

4. 38  454  487  -  10 
4.68  557  487-10 


21 


TABLE  III, 

THE  LOGAEITHMS 

TRIGONOMETRIC    FUNCTIONS : 

Prom  0°  to  0°  3',  or  89°  57'  to  90°,  for  every  second ; 

Prom  0°  to  2°,  or  88°  to  90°,  for  every  ten  seconds ; 

From  1°  to  89°,  for  every  minute. 

Note.  To 

log  sin 

all  the  logarithms  —10  is  to  be  appended. 

no              leg 

U                       log 

tan  =  log  sin 
cos  =  10. 00  000 

ft 

0 

0' 

1' 

2' 

ft 

tf 

0' 

1' 

2' 

f  r 

— 

6.46  373 

6.  76  476 

60 

30 

6.  16  270 

6.  63  982 

6.  86  167 

30 

1 

4.68557 

6. 47  090 

6.  76  836 

59 

31 

6. 17  694 

6.  64  462 

6.86  455 

29 

2 

4.  98  660 

6.  47  797 

6.  77  193 

58 

32 

6. 19  072 

6.  64  936 

6.  86  742 

28 

3 

5. 16  270 

6.  48  492 

6.  77  548 

57 

33 

6.  20  409 

6.  65  406 

6.  87  027 

27 

4 

5. 28  763 

6.49  175 

6.  77  900 

56 

34 

6.  21  705 

6.  65  870 

6.  87  310 

26 

5 

5.  3S  454 

6. 49  S49 

6.  78  248 

55 

35 

6.  22  964 

6.  66  330 

6.  87  591 

25 

6 

5.  46  373 

6.  50  512 

6.  78  595 

54 

36 

6.  24  188 

6.  66  785 

6.  87  870 

24 

7 

5.  53  067 

6.51  165 

6.  78  938 

53 

37 

6.  25  378 

6.  67  235 

6.  88  147 

23 

8 

5. 5S  866 

6.  51  808 

6.  79  278 

52 

38 

6.  26  536 

6.  67  680 

6.  88  423 

22 

9 

5.63  982 

6.  52  442 

6.  79  616 

51 

39 

6.  27  664 

6.  68  121 

6.  88  697 

21 

10 

5.68  557 

6. 53  067 

6.  79  952 

50 

40 

6.  28  763 

6.  68  557 

6.  88  969 

20 

11 

5.  72  697 

6. 53  683 

6.  80  285 

49 

41 

6.  29  836 

6.  68  990 

6.  89  240 

19 

12 

5.76  476 

6. 54  291 

6.  80  615 

48 

42 

6.  30  882 

6.  69  418 

6.  89  509 

18 

13 

5.  79  952 

6.  54  890 

6.  80  943 

47 

43 

6.  31  904 

6.  69  841 

6.  89  776 

17 

14 

5.83  170 

6.  55  481 

6.  81  268 

46 

44 

6.  32  903 

6.  70  261 

6.90  042 

16 

15 

5.  86  167 

6.  56  064 

6.  81  591 

45 

45 

6.  33  879 

6.  70  676 

6.  90  306 

15 

16 

5.  88  969 

6.  56  639 

6.  81  911 

44 

46 

6.  34  833 

6.  71  088 

6.  90  568 

14 

17 

5.  91  602 

6.  57  207 

6.  82  230 

43 

47 

6.  35  767 

6.  71  496 

6.  90  829 

13 

18 

5.94  085 

6.57  767 

6.  82  545 

42 

48 

6.  36  682 

6.  71  900 

6.  91  088 

12 

19 

5. 96  433 

6.  58  320 

6. 82  859 

41 

49 

6.  37  577 

6.  72  300 

6.  91  346 

11 

20 

5.  98  660 

6.  58  866 

6.  83  170 

40 

50 

6.  38  454 

6.  72  697 

6.  91  602 

10 

21 

6. 00  779 

6.  59  406 

6.  83  479 

39 

51 

6.39  315 

6.  73  090 

6. 91  857 

9 

22 

6.  02  800 

6.  59  939 

6.  83  786 

38 

52 

6.  40  158 

6.  73  479 

6. 92  110 

8 

23 

6.  04  730 

6.  60  465 

6.  84  091 

37 

53 

6. 40  985 

6.  73  865 

6. 92  362 

7 

24 

6.  06  579 

6.  60  985 

6.  84  394 

36 

54 

6. 41  797 

6.  74  248 

6. 92  612 

6 

25 

6.08  351 

6.  61  499 

6.  84  694 

35 

55 

6. 42  594 

6.  74  627 

6.  92  861 

5 

26 

6. 10  055 

6.  62  007 

6.  84  993 

34 

56 

6.43  376 

6.  75  003 

6.93  109 

4 

27 

6. 11  694 

6.  62  509 

6.  85  289 

33 

57 

6. 44  145 

6.  75  376 

6.  93  355 

3 

28 

6. 13  273 

6.  63  006 

6.  85  584 

32 

58 

6.  44  900 

6.  75  746 

6. 93  599 

2 

29 

6. 14  797 

6.  63  496 

6.  85  876 

31 

59 

6. 45  643 

6.  76  112 

6. 93  843 

1 

30 

6. 16  270 

6.  63  982 

6.  86  167 

30 

60 

6. 46  373 

6.  76  476 

6.  94  085 

0 

ft 

59' 

58' 

57' 

ft 

ft 

59' 

58' 

57' 

ft 

log  cot  =  log  COS 
log  sin  =  10,  00  000 


89 


log  cos 


22 

0° 

t  tt 

log  sin 

log  tan 

log  COS 

tt  r 

t  tt 

log  sin 

log  tan 

log  cos 

ft    t 

0  0 

_ 



10.00000 

0  60 

10  0 

7.  46  373 

7.46  373 

10.00000 

0  50 

10 

5.68  557 

5.68  557 

10.00000 

50 

10 

7. 47  090 

7.  47  091 

10.00000 

50 

20 

5.  98  660 

5.98  660 

10.00000 

40 

20 

7. 47  797 

7.  47  797 

10.00000 

40 

30 

6. 16  270 

6. 16  270 

10.00000 

30 

30 

7.  48  491 

7.  48  492 

10.00000 

30 

40 

6.  28  763 

6.  28  763 

10.00000 

20 

40 

7.  49  175 

7.  49  176 

10.00000 

20 

50 

6.  38  454 

6. 38  454 

10.00000 

10 

50 

7.  49  849 

7.  49  849 

10.00000 

10 

1  0 

6.46  373 

6.  46  373 

10.00000 

0  59 

11  0 

7.50  512 

7.50  512 

10.00000 

0  49 

10 

6.  53  067 

6.  53  067 

10.00000 

50 

10 

7.  51 165 

7.  51 165 

10.00000 

50 

20 

6.  58  866 

6.  58  866 

10.00000 

40 

20 

7.  51  808 

7.  51  809 

10.00000 

40 

30 

6.  63  982 

6.  63  982 

10.00000 

30 

30 

7.  52  442 

7.  52  443 

10.00000 

30 

40 

6. 68  557 

6.  68  557 

10.00000 

20 

40 

7.  53  067 

7.  53  067 

10.00000 

20 

50 

6.  72  697 

6.  72  697 

10.00000 

10 

50 

7.  53  683 

7.  53  683 

10.00000 

10 

2  0 

6.76476 

6.  76  476 

10.00000 

0  58 

12  0 

7.  54  291 

7.  54  291 

10.00000 

0  48 

10 

6.  79  952 

6.  79  952 

10.00000 

50 

10 

7.  54  890 

7.  54  890 

10.00000 

50 

20 

6. 83  170 

6.  83  170 

10.00000 

40 

20 

7.  55  481 

7.  55  481 

10.00000 

40 

30 

6.  86 167 

6.  86  167 

10.00000 

30 

30 

7.  56  064 

7.  56  064 

10.00000 

30 

40 

6.  88  969 

6.  88  969 

10.00000 

20 

40 

7.  56  639 

7.  56  639 

10.00000 

20 

50 

6. 91  602 

6.  91  602 

10.00000 

10 

50 

7.  57  206 

7.  57  207 

10.00000 

10 

3  0 

6.  94  085 

6.  94  0S5 

10.00000 

0  57 

13  0 

7.  57  767 

7.  57  767 

10.00000 

0  47 

10 

6.  96  433 

6.  96  433 

10.00000 

50 

10 

7.  58  320 

7.58  320 

10.00000 

50 

20 

6.98660 

6.  98  661 

10.00000 

40 

20 

7.  58  866 

7.  58  867 

10.00000 

40 

30 

7.  00  779 

7.  00  779 

10.00000 

30 

30 

7.59406 

7.  59  406 

10.00000 

30 

40 

7.  02  800 

7. 02  800 

10.00000 

20 

40 

7.  59  939 

7.59  939 

10.00000 

20 

50 

7.  04  730 

7.  04  730 

10.00000 

10 

50 

7.60465 

7.  60  466 

10.00000 

10 

4  0 

7. 06  579 

7. 06  579 

10.00000 

0  56 

14  0 

7.  60  985 

7.  60  986 

10.00000 

0  46 

10 

7.08  351 

7.08  352 

10.00000 

50 

10 

7.  61  499 

7.  61  500 

10.00000 

50 

20 

7.10055 

7.10  055 

10.00000 

40 

20 

7.  62  007 

7.  62  008 

10.00000 

40 

30 

7.11694 

7. 11  694 

10.00000 

30 

30 

7.  62  509 

7.  62  510 

10.00000 

30 

40 

7. 13  273 

7. 13  273 

10.00000 

20 

40 

7.  63  006 

7.  63  006 

10.00000 

20 

50 

7. 14  797 

7. 14  797 

10.00000 

10 

50 

7.  63  496 

7.  63  497 

10.00000 

10 

5  0 

7. 16  270 

7. 16  270 

10.00000 

0  55 

15  0 

7.  63  982 

7.  63  982 

10.00000 

0  45 

10 

7.17  694 

7.17  694 

10.00000 

50 

10 

7.  64  461 

7.64  462 

10.00000 

50 

20 

7.19072 

7.19  073 

10.00000 

40 

20 

7.  64  936 

7.  64  937 

10.00000 

40 

30 

7.  20  409 

7.  20  409 

10.00000 

30 

30 

7.  65  406 

7.  65  406 

10.00000 

30 

40 

7.  21  705 

7.  21  705 

10.00000 

20 

40 

7.  65  870 

7.  65  871 

10.00000 

20 

50 

7.  22  964 

7.  22  964 

10.00000 

10 

50 

7.  66  330 

7.  66  330 

10.00000 

10 

6  0 

7.  24  188 

7.  24  188 

10.00000 

0  54 

16  0 

7.  66  784 

7.66  785 

10.00000 

0  44 

10 

7.  25  378 

7.  25  378 

10.00000 

50 

10 

7.  67  235 

7.  67  235 

10.00000 

50 

20 

7.  26  536 

7.  26  536 

10.00000 

40 

20 

7.  67  680 

7.  67  680 

10.00000 

40 

30 

7.  27  664 

7.  27  664 

10.00000 

30 

30 

7.  68  121 

7.  68  121 

10.00000 

30 

40 

7.  28  763 

7.  28  764 

10.00000 

20 

40 

7.  68  557 

7.  68  558 

9.99999 

20 

50 

7.  29  836 

7.  29  836 

10.00000 

10 

50 

7.  68  989 

7.  68  990 

9.99999 

10 

7  0 

7.  30  882 

7.  30  8S2 

10.00000 

0  53 

17  0 

7.69  417 

7. 69  418 

9. 99  999 

0  43 

10 

7.  31  904 

7.  31  904 

10.00000 

50 

10 

7.  69  841 

7.  69  842 

9.  99  999 

50 

20 

7.  32  903 

7.  32  903 

10.00000 

40 

20 

7.  70  261 

7.  70  261 

9.  99  999 

40 

30 

7.  33  879 

7.  33  879 

10.00000 

30 

30 

7.  70  676 

7.  70  677 

9.  99  999 

30 

40 

7.  34  833 

7.  34  833 

10.00000 

20 

40 

7.  71  088 

7.  71  088 

9. 99  999 

20 

50 

7. 35  767 

7.  35  767 

10.00000 

10 

50 

7.  71  496 

7.  71  496 

9.99  999 

10 

8  0 

7.  36  682 

7.  36  682 

10.00000 

0  52 

18  0 

7.  71  900 

7.71900 

9.  99  999 

0  42 

10 

7.37  577 

7.37  577 

10.00000 

50 

10 

7.  72  300 

7.  72  301 

9.  99  999 

50 

20 

7.  38  454 

7.38  455 

10.00000 

40 

20 

7.  72  697 

7.  72  697 

9.  99  999 

40 

30 

7.39  314 

7.39  315 

10.00000 

30 

30 

7.  73  090 

7.  73  090 

9.99  999 

30 

40 

7.40158 

7.  40  158 

10.00000 

20 

40 

7.  73  479 

7.  73  480 

9.  99  999 

20 

50 

7.  40  985 

7.  40  985 

10.00000 

10 

50 

7.  73  865 

7.  73  S66 

9.  99  999 

10 

9  0 

7.41797 

7. 41  797 

10.00000 

0  51 

19  0 

7.  74  248 

7.  74  248 

9.99  999 

0  41 

10 

7. 42  594 

7.  42  594 

10.00000 

50 

10 

7.  74  627 

7.  74  628 

9.  99  999 

50 

20 

7.43  376 

7.  43  376 

10.00000 

40 

20 

7.  75  003 

7.75  004 

9.  99  999 

40 

30 

7. 44  145 

7.  44  145 

10.00000 

30 

30 

7.  75  376 

7.  75  377 

9.99  999 

30 

40 

7. 44  900 

7.44  900 

10.00000 

20 

40 

7.  76  745 

7.  75  746 

9.99  999 

20 

50 

7.  45  643 

7.45  643. 

10.00000 

10 

50 

7.76112 

7.  76  113 

9.99  999 

10 

10  0 

7.  46  373 

7.46  373 

10.00000 

0  50 

20  0 

7.  76  475 

7.76476 

9. 99  999 

0  40 

t  tt 

log  cos 

log  cot 

log  sin 

tt  t 

f    tt 

log  cos 

log  cot 

log  sin 

ft   f 

89 


0° 

23 

t  rr 

log  sin 

log  tan 

log  COS 

tt  r 

t  tf 

log  sin 

log  tan 

log  cos 

ft  t 

20  0 

7.  76  475 

7.76476 

9.99  999 

0  40 

30  0 

7.  94  084 

7.94  086 

9. 99  998 

0  30 

10 

7.  76  836 

7.76  837 

9.  99  999 

50 

10 

7.  94  325 

7.94326 

9.  99  998 

50 

20 

7.  77  193 

7.  77  194 

9.  99  999 

40 

20 

7.  94  564 

7.94  566 

9. 99  998 

40 

30 

7.  77  548 

7.  77  549 

9.  99  999 

30 

30 

7.  94  802 

7. 94  804 

9.  99  998 

30 

40 

7.  77  899 

7.  77  900 

9.99  999 

20 

40 

7.  95  039 

7. 95  040 

9.  99  998 

20 

50 

7.  78  248 

7.  78  249 

9.  99  999 

10 

50 

7. 95  274 

7. 95  276 

9.99998 

10 

21  0 

7.  78  594 

7.  78  595 

9. 99  999 

0  39 

31  0 

7.  95  508 

7.95  510 

9.99998 

0  29 

10 

7.  78  938 

7.78  938 

9.  99  999 

50 

10 

7. 95  741 

7.  95  743 

9.  99  998 

50 

20 

7.  79  278 

7.  79  279 

9.99  999 

40 

20 

7. 95  973 

7. 95  974 

9. 99  998 

40 

30 

7.79  616 

7.79  617 

9.  99  999 

30 

30 

7.  96  203 

7. 96  205 

9.  99  998 

30 

40 

7.  79  952 

7.  79  952 

9.  99  999 

20 

40 

7.96432 

7. 96  434 

9. 99  998 

20 

50 

7.  80  284 

7.  80  285 

9.  99  999 

10 

50 

7.  96  660 

7.  96  662 

9.99998 

10 

22  0 

7.  80  615 

7.80  615 

9.99  999 

0  38 

32  0 

7.  96  887 

7.  96  889 

9. 99  998 

0  28 

10 

7.  80  942 

7.  80  943 

9.  99  999 

50 

10 

7.97113 

7.97114 

9. 99  998 

50 

20 

7.  81  268 

7.  81  269 

9.  99  999 

40 

20 

7.97  337 

7. 97  339 

9.  99  998 

40 

30 

7.  81  591 

7.  81  591 

9.  99  999 

30 

30 

7.  97  560 

7.  97  562 

9.  99  998 

30 

40 

7.81911 

7.81912 

9.  99  999 

20 

40 

7. 97  782 

7.  97  784 

9. 99  998 

20 

50 

7.  82  229 

7.  82  230 

9. 99  999 

10 

50 

7.  98  003 

7.  98  005 

9. 99  998 

10 

23  0 

7.  82  545 

7.  82  546 

9.  99  999 

0  37 

33  0 

7.  98  223 

7. 98  225 

9.  99  998 

0  27 

10 

7.  82  859 

7.  82  860 

9. 99  999 

50 

10 

7.  98  442 

7.  98  444 

9. 99  998 

50 

20 

7.  83  170 

7.  83  171 

9.  99  999 

40 

20 

7.  98  660 

7. 98  662 

9.99998 

40 

30 

7.  83  479 

7.  83  480 

9.  99  999 

30 

30 

7.  98  876 

7.98  878 

9.  99  998 

30 

40 

7.  83  786 

7.  83  787 

9.  99  999 

20 

40 

7.  99  092 

7.  99  094 

9.  99  998 

20 

50 

7.  84  091 

7. 84  092 

9. 99  999 

10 

50 

7.  99  306 

7.  99  308 

9. 99  998 

10 

24  0 

7.  84  393 

7. 84  394 

9.  99  999 

0  36 

34  0 

7.  99  520 

7.  99  522 

9.  99  998 

0  26 

10 

7.  84  69+ 

7.  84  695 

9.  99  999 

50 

10 

7.  99  732 

7.  99  734 

9.  99  998 

50 

20 

7.  84  992 

7.84  994 

9.  99  999 

40 

20 

7.  99  943 

7.  99  946 

9.  99  998 

40 

30 

7.  85  289 

7.  85  290 

9. 99  999 

30 

30 

8.  00  154 

8.  00  156 

9. 99  998 

30 

40 

7.  85  583 

7.  85  584 

9.  99  999 

20 

40 

8.  00  363 

8.  00  365 

9. 99  998 

20 

50 

7.  85  876 

7.85  877 

9. 99  999 

10 

50 

8.  00  571 

8.00  574 

9.  99  998 

10 

25  0 

7.  86  166 

7. 86  167 

9. 99  999 

0  35 

35  0 

8.  00  779 

8.  00  781 

9.  99  998 

0  25 

10 

7.  86  455 

7.  86  456 

9.  99  999 

50 

10 

8.  00  985 

8.  00  987 

9.  99  998 

50 

20 

7.  86  741 

7.  86  743 

9.  99  999 

40 

20 

8.  01  190 

8.  01 193 

9. 99  998 

40 

30 

7.  87  026 

7.  87  027 

9.  99  999 

30 

30 

8.  01  395 

8.  01  397 

9.99998 

30 

40 

7.  87  309 

7.87  310 

9.  99  999 

20 

40 

8.  01  598 

8.  01  600 

9.99998 

20 

50 

7.  87  590 

7. 87  591 

9.  99  999 

10 

50 

8.  01  801 

8.  01  803 

9.99998 

10 

26  0 

7.  87  870 

7.  87  871 

9.  99  999 

0  34 

36  0 

8. 02  002 

8.  02  004 

9. 99  998 

0  24 

10 

7.  88  147 

7.  88  148 

9.  99  999 

50 

10 

8.  02  203 

8.  02  205 

9.  99  998 

50 

20 

7.  88  423 

7.  88  424 

9.  99  999 

40 

20 

8.  02  402 

8.  02  405 

9.  99  998 

40 

30 

7.  88  697 

7. 88  698 

9.  99  999 

30 

30 

8. 02  601 

8.  02  604 

9. 99  998 

30 

40 

7.  88  969 

7.  88  970 

9.  99  999 

20 

40 

8.  02  799 

8.  02  801 

9. 99  998 

20 

50 

7. 89  240 

7.  89  241 

9.  99  999 

10 

50 

8.  02  996 

8.  02  998 

9.  99  998 

10 

27  0 

7.  89  509 

7.  89  510 

9.99  999 

0  33 

37  0 

8.  03  192 

8.  03  194 

9.  99  997 

0  23 

•  10 

7. 89  776 

7.  89  777 

9.  99  999 

50 

10 

8.  03  387 

8. 03  390 

9.  99  997 

50 

20 

7. 90  041 

7.  90  043 

9.  99  999 

40 

20 

8.  03  581 

8.  03  584 

9.  99  997 

40 

30 

7. 90  305 

7.  90  307 

9.  99  999 

30 

30 

8. 03  775 

8.  03  777 

9. 99  997 

30 

40 

7. 90  568 

7. 90  569 

9.  99  999 

20 

40 

8.  03  967 

8.  03  970 

9. 99  997 

20 

50 

7. 90  829 

7.  90  830 

9.  99  999 

10 

50 

8.04159 

8. 04  162 

9. 99  997 

10 

28  0 

7. 91  088 

7. 91  0S9 

9.  99  999 

0  32 

38  0 

8. 04  350 

8.04  353 

9. 99  997 

0  22 

10 

7. 91  346 

7.91347 

9.  99  999 

50 

10 

8.  04  540 

8. 04  543 

9. 99  997 

50 

20 

7.  91  602 

7.  91  603 

9. 99  999 

40 

20 

8. 04  729 

8.  04  732 

9.  99  997 

40 

30 

7.  91  857 

7.  91  858 

9.  99  999 

30 

30 

8.  04  918 

8.04  921 

9.  99  997 

30 

40 

7.92  110 

7.92  111 

9. 99  998 

20 

40 

8.  05  105 

8.  05  108 

9. 99  997 

20 

50 

7. 92  362 

7. 92  363 

9. 99  998 

10 

50 

8.  05  292 

8.  05  295 

9. 99  997 

10 

29  0 

7.  92  612 

7.  92  613 

9.  99  998 

0  31 

39  0 

8. 05  478 

8.  05  481 

9. 99  997 

0  21 

10 

7. 92  861 

7. 92  862 

9.  99  998 

50 

10 

8.  05  663 

8.  05  666 

9.  99  997 

50 

20 

7. 93  108 

7.  93  110 

9.  99  998 

40 

20 

8. 05  848 

8. 05  851 

9.99  997 

40 

30 

7.  93  354 

7. 93  356 

9. 99  998 

30 

30 

8.  06  031 

8.  06  034 

9. 99  997 

30 

40 

7. 93  599 

7.  93  601 

9. 99  998 

20 

40 

8.  06  214 

8.06  217 

9.  99  997 

20 

50 

7. 93  842 

7.  93  844 

9.  99  998 

10 

50 

8. 06  396 

8. 06  399 

9.  99  997 

10 

30  0 

7.  94  084 

7. 94  086 

9.99  998 

0  30 

40  0 

8.06  578 

8.  06  581 

9.99997 

0  20 

f   ft 

log  cos 

log  cot 

log  sin 

ft  t 

/  tf 

log  cos 

log  cot 

log  sin 

"' 

89 


24 

0° 

r  ft 

log  sin 

log  tan 

log  COS 

rr  t 

r  rr 

log  sin 

log  tan 

log  cos 

rr  r 

40  0 

8.06  578 

8.  06  581 

9.99997 

0  20 

50  0 

8.  16  26S 

8. 16  273 

9.  99  995 

OlO 

10 

8.  06  758 

8.  06  761 

9.  99  997 

50 

10 

8.16413 

8.16  417 

9.  99  995 

50 

20 

8. 06  938 

8.  06  941 

9.  99  997 

40 

20 

8.16  557 

8.16  561 

9.99  995 

40 

30 

8.07117 

8.  07  120 

9.  99  997 

30 

30 

8. 16  700 

8.  16  705 

9.  99  995 

30 

40 

8. 07  295 

8. 07  299 

9.  99  997 

20 

40 

8. 16  843 

8. 16  848 

9.  99  995 

20 

50 

8.  07  473 

8.  07  476 

9.99997 

10 

50 

8. 16  986 

8.16  991 

9.  99  995 

10 

41  0 

8.  07  650 

8. 07  653 

9. 99  997 

0  19 

51  0 

8.17128 

8. 17  133 

9.  99  995 

0  9 

10 

8. 07  826 

8.  07  829 

9.  99  997 

50 

10 

8.17  270 

8.17  275 

9.  99  995 

50 

20 

8.  08  002 

8. 08  005 

9. 99  997 

40 

20 

8.17  411 

8.17  416 

9.  99  995 

40 

30 

8.  08  176 

8.  08  180 

9.  99  997 

30 

30 

8.  17  552 

8.17  557 

9.  99  995 

30 

40 

8.  08  350 

8.  08  354 

9.  99  997 

20 

40 

8.  17  692 

8. 17  697 

9.  99  995 

20 

50 

8.  08  524 

8. 08  527 

9.  99  997 

10 

50 

8. 17  832 

8. 17  837 

9.  99  995 

10 

42  0 

8.  08  696 

8. 08  700 

9.  99  997 

0  18 

52  0 

8.17  971 

8.17  976 

9.  99  995 

0  8 

10 

8. 08  868 

8.  08  872 

9.  99  997 

50 

10 

8.18110 

8. 18  115 

9.  99  995 

50 

20 

8.  09  040 

8. 09  043 

9. 99  997 

40 

20 

8.  18  249 

8. 18  254 

9. 99  995 

40 

30 

8.  09  210 

8. 09  214 

9.  99  997 

30 

30 

8. 18  387 

8. 18  392 

9.  99  995 

30 

40 

8.  09  380 

8. 09  384 

9.  99  997 

20 

40 

8. 18  524 

8. 18  530 

9.  99  995 

20 

50 

8.  09  550 

8. 09  553 

9.  99  997 

10 

50 

8.  18  662 

8. 18  667 

9.99  995 

10 

43  0 

8. 09  718 

8. 09  722 

9.99  997 

0  17 

53  0 

8. 18  798 

8. 18  804 

9.99  995 

0  7 

10 

8.  09  886 

8.  09  890 

9.  99  997 

50 

10 

8. 18  935 

8. 18  940 

9.  99  995 

50 

20 

8.10  054 

8.10  057 

9.  99  997 

40 

20 

8.19  071 

8.19  076 

9.  99  995 

40 

30 

8. 10  220 

8. 10  224 

9. 99  997 

30 

30 

8. 19  206 

8.19  212 

9. 99  995 

30 

40 

8. 10  386 

8. 10  390 

9. 99  997 

20 

40 

8. 19  341 

8.19  347 

9.  99  995 

20 

50 

8.10  552 

8. 10  555 

9. 99  996 

10 

50 

8. 19  476 

8. 19  481 

9.  99  995 

10 

44  0 

8.10  717 

8. 10  720 

9. 99  996 

0  16 

54  0 

8. 19  610 

8. 19  616 

9. 99  995 

0  6 

10 

8.10  881 

8.  10  884 

9. 99  996 

50 

10 

8. 19  744 

8.  19  749 

9.  99  995 

50 

20 

8.11044 

8. 11  048 

9.  99  996 

40 

20 

8.19  877 

8.  19  883 

9.  99  995 

40 

30 

8. 11  207 

8.11211 

9.  99  996 

30 

30 

8.  20  010 

8.  20  016 

9.  99  995 

30 

40 

8. 11  370 

8. 11  373 

9. 99  996 

20 

40 

8.  20  143 

8.  20  149 

9.  99  995 

20 

50 

8. 11  531 

8. 11  535 

9. 99  996 

10 

50 

8.  20  275 

8.  20  281 

9.  99  994 

10 

45  0 

8.11693 

8. 11  696 

9.  99  996 

0  15 

55  0 

8.  20  407 

8.20413 

9.99  994 

0  5 

10 

8. 11  853 

8.11857 

9.  99  996 

50 

10 

8.  20  538 

8.  20  544 

9.  99  994 

50 

20 

8.12  013 

8.12  017 

9.  99  996 

40 

20 

8.  20  669 

8.  20  675 

9.  99  994 

40 

30 

8.12  172 

8. 12  176 

9. 99  996 

30 

30 

8.  20  800 

8.  20  806 

9.  99  994 

30 

40 

8.12  331 

8. 12  335 

9.  99  996 

20 

40 

8.  20  930 

8.  20  936 

9.  99  994 

20 

50 

8. 12  489 

8. 12  493 

9. 99  996 

10 

50 

8.  21  060 

8.  21  066 

9.  99  994 

10 

46  0 

8.12  647 

8.12  651 

9.  99  996 

0  14 

56  0 

8.  21 189 

8.  21 195 

9.  99  994 

0  4 

10 

8. 12  804 

8. 12  808 

9.  99  996 

50 

10 

8.  21  319 

8.  21  324 

9.  99  994 

50 

20 

8.12  961 

8. 12  965 

9.  99  996 

40 

20 

8.  21  447 

8.  21  453 

9.  99  994 

40 

30 

8. 13  117 

8. 13  121 

9.  99  996 

30 

30 

8.  21  576 

8.  21  581 

9.  99  994 

30 

40 

8.13  272 

8. 13  276 

9. 99  996 

20 

40 

8.  21  703 

S.  21  709 

9.99  944 

20 

50 

8. 13  427 

8. 13  431 

9. 99  996 

10 

50 

8.  21  831 

8.  21  837 

9.  99  994 

10 

47  0 

8. 13  581 

8. 13  585 

9.  99  996 

0  13 

57  0 

8.  21  958 

8.  21  964 

9.99  994 

0  3 

10 

8. 13  735 

8. 13  739 

9.  99  996 

50 

10 

8.  22  085 

8.  22  091 

9.  99  994 

50 

20 

8. 13  888 

8. 13  892 

9.  99  996 

40 

20 

8.22  211 

8.22  217 

9.  99  994 

40 

30 

8. 14  041 

8. 14  045 

9.  99  996 

30 

30 

8.22  337 

8.  22  343 

9.  99  994 

30 

40 

8. 14 193 

8. 14  197 

9.  99  996 

20 

40 

8.  22  463 

8.  22  469 

9.  99  994 

20 

50 

8. 14  344 

8. 14  348 

9.  99  996 

10 

50 

8.  22  588 

8.  22  595 

9.99  994 

10 

48  0 

8. 14  495 

8. 14  500 

9.  99  996 

0  12 

58  0 

8.  22  713 

8.  22  720 

9. 99  994 

0  2 

10 

8. 14  646 

8. 14  650 

9.  99  996 

50 

10 

8.  22  838 

S.  22  844 

9.  99  994 

50 

20 

8. 14  796 

8. 14  800 

9.  99  996 

40 

20 

8.  22  962 

8.  22  96S 

9.99  994 

40 

30 

8. 14  945 

8. 14  950 

9.  99  996 

30 

30 

8.  23  0S6 

8.  23  092 

9.  99  994 

30 

40 

8. 15  094 

8.  15  099 

9. 99  996 

20 

40 

8.  23  210 

8.  23  216 

9.  99  994 

20 

50 

8.15  243 

8. 15  247 

9.  99  996 

10 

50 

8.  23  333 

8.  23  339 

9.  99  994 

10 

49  0 

8. 15  391 

8. 15  395 

9.  99  996 

0  11 

59  0 

8.  23  456 

8.  23  462 

9.  99  994 

0  1 

10 

8. 15  538 

8. 15  543 

9.  99  996 

50 

10 

8.  23  578 

8.  23  585 

9.99  994 

50 

20 

8. 15  685 

8. 15  690 

9. 99  996 

40 

20 

8.  23  700 

8.  23  707 

9.99  994 

40 

30 

8. 15  832 

8. 15  836 

9.  99  996 

30 

30 

8.  23  822 

8.  23  829 

9.  99  993 

30 

40 

8. 15  978 

8. 15  982 

9.  99  995 

20 

40 

8.  23  944 

8.  23  950 

9.  99  993 

20 

50 

8. 16  123 

8. 16  128 

9.  99  995 

10 

50 

8.  24  065 

8.  24  071 

9.  99  993 

10 

50  0 

8. 16  268 

8. 16  273 

9. 99  995 

0  10 

60  0 

8.  24  186 

8.  24  192 

9.99  993 

oo 

r  rr 

log  cos 

log  cot 

log  sin 

rr  t 

/  rr 

log  cos 

log  cot 

log  sin 

• 

89 


r 

25 

t  tt 

log  sin 

log  tan 

log  cos 

ft   t 

t  tt 

log  sin 

log  tan 

log  COS 

tt  t 

0  0 

8.  24 186 

8.  24  192 

9.  99  993 

0  60 

10  o 

8.  30  879 

8.  30  888 

9.  99  991 

0  50 

10 

8.  24  306 

8.24  313 

9.  99  993 

50 

10 

8.  30  983 

8.  30  992 

9.  99  991 

50 

20 

8.  24  426 

8.  24  433 

9.  99  993 

40 

20 

8.  31  086 

8.  31  095 

9.  99  991 

40 

30 

8.  24  546 

8.  24  553 

9.  99  993 

30 

30 

8. 31 188 

8.  31 198 

9.99  991 

30 

40 

8. 24  665 

8.  24  672 

9.  99  993 

20 

40 

8.  31  291 

8.  31  300 

9.  99  991 

20 

50 

8.  24  785 

8. 24  791 

9.  99  993 

10 

50 

8.  31  393 

8.  31  403 

9.99991 

10 

1  0 

8.  24  903 

8.  24  910 

9. 99  993 

0  59 

11  0 

8. 31  495 

8. 31  505. 

9.  99  991 

0  49 

10 

8.  25  022 

8.  25  029 

9.  99  993 

50 

10 

8.  31  597 

8.  31  606 

9. 99  991 

50 

20 

8.  25  140 

8.  25  147 

9.  99  993 

40 

20 

8.  31  699 

8.  31  708 

9.  99  991 

40 

30 

8.  25  258 

8.  25  265 

9.  99  993 

30 

30 

8.  31  800 

8.  31  809 

9. 99  991 

30 

40 

8.25  375 

8.  25  382 

9.  99  993 

20 

40 

8.  31  901 

8.31911 

9.  99  991 

20 

50 

8.  25  493 

8.  25  500 

9.  99  993 

10 

50 

8.  32  002 

8.32  012 

9. 99  991 

10 

2  0 

8.  25  609 

8.  25  616 

9.  99  993 

0  58 

12  0 

8.  32  103 

8.32112 

9.  99  990 

0  48 

10 

8.  25  726 

8.  25  733 

9. 99  993 

50 

10 

8.  32  203 

8.32  213 

9. 99  990 

50 

20 

8.  25  842 

8.  25  849 

9. 99  993 

40 

20 

8.  32  303 

8.32  313 

9.  99  990 

40 

30 

8.  25  958 

8.  25  965 

9.  99  993 

30 

30 

8.  32  403 

8.32  413 

9. 99  990 

30 

40 

8.26074 

8.  26  081 

9. 99  993 

20 

40 

8.  32  503 

8.32  513 

9. 99  990 

20 

50 

8.  26  189 

8. 26  196 

9.  99  993 

10 

50 

8.  32  602 

8.  32  612 

9.  99  990 

10 

3  0 

8.  26  304 

8.26  312 

9.99  993 

0  57 

13  0 

8.  32  702 

8.32  711 

9.  99  990 

0  47 

10 

8.26  419 

8.26426 

9. 99  993 

50 

10 

8.  32  801 

8.32  811 

9. 99  990 

50 

20 

8.  26  533 

8.  26  541 

9. 99  993 

40 

20 

8.  32  899 

8.  32  909 

9.  99  990 

40 

30 

8.  26  648 

8.  26  655 

9. 99  993 

30 

30 

8.  32  998 

8. 33  008 

9.  99  990 

30 

40 

8.  26  761 

8.  26  769 

9.  99  993 

20 

40 

8.  33  096 

8.  33  106 

9.  99  990 

20 

50 

8.  26  875 

8.  26  8S2 

9.99  993 

10 

50 

8.  33  195 

8.  33  205 

9. 99  990 

10 

4  0 

S.  26  9SS 

8.  26  996 

9.  99  992 

0  56 

14  0 

8.  33  292 

8.  33  302 

9. 99  990 

0  46 

10 

8.27101 

8.  27  109 

9. 99  992 

50 

10 

8.  33  390 

8.  33  400 

9.  99  990 

50 

20 

8.27  214 

8.  27  221 

9.  99  992 

40 

20 

8.  33  488 

8.  33  498 

9.  99  990 

40 

30 

8.  27  326 

8.  27  334 

9. 99  992 

30 

30 

8.  33  585 

8.  33  595 

9.  99  990 

30 

40 

8.  27  438 

8.  27  446 

9. 99  992 

20 

40 

8.  33  682 

8.  33  692 

9.  99  990 

20 

50 

8.  27  550 

8.27  558 

9. 99  992 

10 

50 

8.  33  779 

8.  33  789 

9. 99  990 

10 

5  0 

8.  27  661 

8.  27  669 

9. 99  992 

0  55 

15  0 

8.  33  875 

8.  33  886 

9.99990 

0  45 

10 

8.  27  773 

8.  27  780 

9. 99  992 

50 

10 

8.  33  972 

8. 33  982 

9.  99  990 

50 

20 

8.  27  883 

8.  27  891 

9.  99  992 

40 

20 

8.  34  068 

8. 34  078 

9.  99  990 

40 

30 

8. 27  994 

8.  28  002 

9. 99  992 

30 

30 

8.  34  164 

8.34174 

9.  99  990 

30 

40 

8.  28  104 

8.28112 

9.  99  992 

20 

40 

8.  34  260 

8.  34  270 

9.  99  989 

20 

50 

8.28  215 

8.  28  223 

9. 99  992 

10 

50 

8.  34  355 

8. 34  366 

9.  99  989 

10 

6  0 

8.2S324 

8.  2S  332 

9. 99  992 

0  54 

16  0 

8.34  450 

8. 34  461 

9. 99  989 

0  44 

10 

8.  28  434 

8.  28  442 

9. 99  992 

50 

10 

8.  34  546 

8.  34  556 

9.  99  989 

50 

20 

8.  28  543 

8.28  551 

9. 99  992 

40 

20 

8.  34  640 

8.34  651 

9.  99  989 

40 

30 

8.  28  652 

8.  28  660 

9.  99  992 

30 

30 

8.  34  735 

8.  34  746 

9.  99  989 

30 

40 

8. 28  761 

8.  28  769 

9. 99  992 

20 

40 

8.  34  830 

8. 34  840 

9.  99  989 

20 

50 

8.  28  869 

8.28  877 

9.  99  992 

10 

50 

8. 34  924 

8.  34  935 

9.  99  989 

10 

7  0 

8.28  977 

8.  2S  986 

9.  99  992 

0  53 

17  0 

8.  35  018 

8.  35  029 

9.  99  989 

0  43 

10 

8.  29  085 

8.  29  094 

9. 99  992 

50 

10 

8.35112 

8.35  123 

9. 99  989 

50 

20 

8.  29  193 

8.  29  201 

9. 99  992 

40 

20 

8.  35  206 

8.35  217 

9.  99  989 

40 

30 

8.  29  300 

8.  29  309 

9.  99  992 

30 

30 

8.  35  299 

8.  35  310 

9. 99  989 

30 

40 

8.  29  407 

8.29  416 

9.  99  992 

20 

40 

8.  35  392 

8.  35  403 

9.  99  989 

20 

50 

8.29  514 

8.  29  523 

9.  99  992 

10 

50 

8. 35  485 

8.  35  497 

9.  99  989 

10 

8  0 

8.29621 

8.  29  629 

9. 99  992 

0  52 

18  0 

8.  35  578 

8.  35  590 

9.  99  989 

0  42 

10 

8.  29  727 

8.  29  736 

9. 99  991 

50 

10 

8.  35  671 

8.  35  682 

9. 99  989 

50 

20 

8.29833 

8.  29  842 

9. 99  991 

40 

20 

8.  35  764 

8.  35  775 

9.99  989 

40 

30 

8.29939 

8.29947 

9.  99  991 

30 

30 

8.  35  856 

8.  35  867 

9.  99  989 

30 

40 

8.30044 

8.30053 

9.  99  991 

20 

40 

8.  35  948 

8.  35  959 

9.  99  989 

20 

50 

8.  30  150 

8.  30  158 

9.  99  991 

10 

50 

8.  36  040 

8.36051 

9. 99  989 

10 

9  0 

8. 30  255 

8. 30  263 

9.99  991 

0  51 

19  0 

8.36131 

8.  36  143 

9.  99  989 

0  41 

10 

8.30  359 

8. 30  368 

9.  99  991 

50 

10 

8.  36  223 

8.  36  235 

9.  99  988 

50 

20 

8.30464 

8.  30  473 

9. 99  991 

40 

20 

8.36314 

8. 36  326 

9.  99  988 

40 

30 

8. 30  568 

8.30  577 

9.  99  991 

30 

30 

8.  36  405 

8.36  417 

9. 99  988 

30 

40 

8.30  672 

8.  30  681 

9.  99  991 

20 

40 

8.  36  496 

8.  36  508 

9.  99  988 

20 

50 

8.  30  776 

8. 30  785 

9. 99  991 

10 

50 

8.36  587 

8.  36  599 

9.  99  988 

10 

10  0 

8. 30  879 

8.  30  888 

9.  99  991 

0  50 

20  0 

8.36  678 

8.  36  689 

9. 99  988 

0  40 

t   tt 

log  cos 

log  cot 

log  sin 

ft  r 

t  tt 

log  cos 

log  cot 

log  sin 

tt  t 

88 


26 

1° 

f  ft 

log  sin 

log  tan 

log  cos 

ft  1 

f  tt 

log  sin 

log  tan 

log  cos 

ft  t 

20  0 

8.  36  678 

8.36  689 

9.99  988 

0  40 

30  0 

8.  41  792 

8.  41  807 

9.  99  985 

0  30 

10 

8.  36  768 

8.  36  780 

9. 99  988 

50 

10 

8.  41  872 

8.  41  887 

9.  99  985 

50 

20 

8.  36  858 

8.  36  870 

9.  99  988 

40 

20 

8. 41  952 

8.  41  967 

9.  99  985 

40 

30 

8.  36  948 

8.  36  960 

9.  99  988 

30 

30 

8. 42  032 

8. 42  048 

9.  99  985 

30 

40 

8.  37  038 

8.37  050 

9.  99  988 

20 

40 

8.42  112 

8. 42  127 

9.  99  985 

20 

50 

8. 37  128 

8.  37  140 

9.  99  988 

10 

50 

8. 42  192 

8.  42  207 

9.  99  985 

10 

21  0 

8.37  217 

8.  37  229 

9.  99  988 

0  39 

31  0 

8.  42  272 

8.  42  287 

9.  99  985 

0  29 

10 

8.  37  306 

8.37  318 

9.  99  988 

50 

10 

8.42  351 

8.  42  366 

9.  99  985 

50 

20 

8.  37  395 

8.  37  408 

9.  99  988 

40 

20 

8.  42  430 

8.  42  446 

9.  99  985 

40 

30 

8.  37  484 

8.  37  497 

9.  99  988 

30 

30 

8.42  510 

8. 42  525 

9.  99  985 

30 

40 

8.37  573 

8.37  585 

9.  99  988 

20 

40 

8.  42  589 

8.  42  604 

9.  99  985 

20 

50 

8.  37  662 

8.  37  674 

9.  99  988 

10 

50 

8. 42  667 

8. 42  683 

9.  99  985 

10 

22  0 

8.  37  750 

8.  37  762 

9.  99  988 

0  38 

32  0 

8.  42  746 

8. 42  762 

9.  99  984 

0  28 

10 

8.  37  838 

8.  37  850 

9.  99  988 

50 

10 

8. 42  825 

8.  42  840 

9.  99  984 

50 

20 

8.  37  926 

8.  37  938 

9.  99  988 

40 

20 

8. 42  903 

8. 42  919 

9.  99  984 

40 

30 

8.38  014 

8.  38  026 

9.  99  987 

30 

30 

8.  42  982 

8.  42  997 

9.  99  984 

30 

40 

8.  38  101 

8.38114 

9.  99  987 

20 

40 

8. 43  060 

8.  43  075 

9.  99  984 

20 

50 

8.  38  189 

8.  38  202 

9.  99  987 

10 

50 

8. 43  138 

8.  43  154 

9.  99  984 

10 

23  0 

8.  38  276 

8.  38  289 

9.  99  987 

0  37 

33  0 

8.43  216 

8.  43  232 

9.99984 

0  27 

10 

8.  38  363 

8.  38  376 

9.  99  987 

50 

10 

8. 43  293 

8. 43  309 

9.  99  984 

50 

20 

8.  38  450 

8.  38  463 

9.  99  987 

40 

20 

8.43  371 

8.  43  387 

9.  99  984 

40 

30 

8.38  537 

8.  38  550 

9.  99  987 

30 

30 

8.  43  448 

8.  43  464 

9.  99  984 

30 

40 

8.  38  624 

8.  38  636 

9.  99  987 

20 

40 

8. 43  526 

8. 43  542 

9. 99  984 

20 

50 

8. 38  710 

8.  38  723 

9.  99  987 

10 

50 

8. 43  603 

8. 43  619 

9.  99  984 

10 

24  0 

8.  38  796 

8.  38  809 

9.  99  987 

0  36 

34  0 

8. 43  680 

8.  43  696 

9. 99  984 

0  26 

10 

8.  38  882 

8.  38  895 

9.  99  987 

50 

10 

8.  43  757 

8.43  773 

9.  99  984 

50 

20 

8.  38  968 

8.  38  981 

9.  99  987 

40 

20 

8. 43  834 

8. 43  850 

9.  99  984 

40 

30 

8.  39  054 

8.  39  067 

9.  99  987 

30 

30 

8.  43  910 

8. 43  927 

9. 99  984 

30 

40 

8.  39  139 

8.  39  153 

9.  99  9S7 

20 

40 

8.  43  987 

8.  44  003 

9.  99  984 

20 

50 

8. 39  225 

8.  39  238 

9.99  987 

10 

50 

8. 44  063 

8. 44  080 

9.  99  983 

10 

25  0 

8.39  310 

8.  39  323 

9.  99  987 

0  35 

35  0 

8. 44  139 

8. 44  156 

9.  99  983 

0  25 

10 

8. 39  395 

8.  39  408 

9.  99  987 

50 

10 

8. 44  216 

8.  44  232 

9.  99  9S3 

50 

20 

8.39480 

8.  39  493 

9.  99  987 

40 

20 

8. 44  292 

8.  44  30S 

9.  99  983 

40 

30 

8.  39  565 

8.  39  578 

9.  99  987 

30 

30 

8. 44  367 

8.  44  3S4 

9. 99  983 

30 

40 

8.  39  649 

8.  39  663 

9.  99  987 

20 

40 

8. 44  443 

8.  44  460 

9.  99  983 

20 

50 

8.  39  734 

8.  39  747 

9. 99  986 

10 

50 

8.44  519 

8. 44  536 

9.  99  983 

10 

26  0 

8.  39  818 

8.  39  832 

9.  99  986 

0  34 

36  0 

8. 44  594 

8.44  611 

9.  99  983 

0  24 

10 

8.  39  902 

8.  39  916 

9.  99  986 

50 

10 

8. 44  669 

8. 44  686 

9.  99  983 

50 

20 

8.  39  986 

8.  40  000 

9.  99  986 

40 

20 

8.  44  745 

8. 44  762 

9.  99  983 

40 

30 

8.  40  070 

8.  40  083 

9.  99  986 

30 

30 

8. 44  820 

8. 44  837 

9. 99  983 

30 

40 

8.  40  153 

8.  40  167 

9.  99  986 

20 

40 

8. 44  895 

8.  44  912 

9.  99  983 

20 

50 

8. 40  237 

8.40  251 

9.  99  986 

10 

50 

8. 44  969 

8. 44  987 

9.  99  983 

10 

27  0 

8.40320 

8.  40  334 

9.  99  986 

0  33 

37  0 

8.45  044 

8. 45  061 

9.  99  9S3 

0  23 

10 

8. 40  403 

8.40  417 

9.  99  986 

50 

10 

8. 45  119 

8. 45  136 

9.  99  983 

50 

20 

8. 40  486 

8.  40  500 

9.  99  986 

40 

20 

8. 45  193 

8.  45  210 

9.  99  983 

40 

30 

8. 40  569 

8.  40  583 

9.  99  986 

30 

30 

8.  45  267 

8. 45  285 

9.  99  9S3 

30 

40 

8.40  651 

8.  40  665 

9.  99  986 

20 

40 

8. 45  341 

8. 45  359 

9. 99  982 

20 

50 

8.  40  734 

8.  40  748 

9.  99  986 

10 

50 

8.45  415 

8.  45  433 

9.  99  9S2 

10 

28  0 

8. 40  816 

8.  40  830 

9.  99  986 

0  32 

38  0 

8. 45  489 

8. 45  507 

9. 99  9S2 

0  22 

10 

8. 40  898 

8.  40  913 

9.  99  986 

50 

10 

8. 45  563 

8.45  581 

9. 99  982 

50 

20 

8.  40  980 

8.  40  995 

9.  99  986 

40 

20 

8. 45  637 

8.45  655 

9.  99  9S2 

40 

30 

8. 41  062 

8.  41  077 

9.  99  986 

30 

30 

8.45  710 

S.  45  728 

9. 99  982 

30 

40 

8. 41 144 

8.  41  158 

9.  99  986 

20 

40 

8. 45  784 

8. 45  802 

9.99982 

20 

50 

8. 41  225 

8.  41  240 

9.  99  986 

10 

50 

8.45  857 

8. 45  875 

9. 99  982 

10 

29  0 

8. 41  307 

8.  41  321 

9.  99  985 

0  31 

39  0 

8. 45  930 

8. 45  948 

9.99982 

0  21 

10 

8. 41  388 

8.  41  403 

9.  99  985 

50 

10 

8.46003 

8.46  021 

9. 99  982 

50 

20 

8. 41  469 

8.  41  484 

9.  99  985 

40 

20 

8.46076 

8.46094 

9.  99  982 

40 

30 

8.41550 

8.  41  565 

9.  99  985 

30 

30 

8. 46  149 

8.  46  167 

9.  99  9S2 

30 

40 

8.  41  631 

8.  41  646 

9.  99  985 

20 

40 

8. 46  222 

8.  46  240 

9.  99  982 

20 

50 

8.  41  711 

8. 41  726 

9.  99  985 

10 

50 

8. 46  294 

8.46  312 

9.  99  982 

10 

30  0 

8. 41  792 

8.  41  807 

9.  99  985 

0  30 

40  0 

8. 46  366 

8.  46  385 

9.  99  982 

0  20 

r   tt 

log  cos 

log  cot 

log  sin 

tt   t 

t   ft 

log  cos 

log  cot 

log  sin 

tt  t 

88' 


1° 

27 

r  tr 

log  sin 

log  tan 

log  COS 

tt   f 

f  tf 

log  sin 

log  tan 

log  cos 

tt  t 

40  0 

8. 46  366 

8.46  385 

9.  99  982 

0  20 

50  0 

8.  50  504 

8. 50  527 

9.99978 

oio 

10 

8.  46  439 

8.46  457 

9.99  982 

50 

10 

8.50  570 

8.  50  593 

9.99978 

60 

20 

8.46  511 

8.  46  529 

9.  99  982 

40 

20 

8.50636 

8.50  658 

9.99978 

40 

30 

8. 46  583 

8.  46  602 

9.99  981 

30 

30 

8.  50  701 

8.  50  724 

9.99  978 

30 

40 

8.46  655 

8.  46  674 

9.  99  981 

20 

40 

8.  50  767 

8.  50  789 

9. 99  977 

20 

50 

8.  46  727 

8.  46  745 

9.99981 

10 

50 

8.50832 

8.  50  855 

9.99977 

10 

41  0 

8.  46  799 

8.46  817 

9. 99  981 

0  19 

51  0 

8.50  897 

8.50920 

9.99977 

0  9 

10 

8. 46  870 

8. 46  8S9 

9.  99  9S1 

50 

10 

8.50963 

8.50985 

9.99977 

50 

20 

S.  46  942 

8.  46  960 

9.  99  981 

40 

20 

8.  51  028 

8.  51  050 

9. 99  977 

40 

30 

8.47  013 

8. 47  032 

9.  99  981 

30 

30 

8.  51  092 

8.51115 

9.99977 

30 

40 

8.  47  084 

8. 47  103 

9.  99  981 

20 

40 

8.51157 

8.  51 180 

9.99977 

20 

50 

8.47155 

8.47174 

9.  99  981 

10 

50 

8.  51  222 

8.51245 

9.99977 

10 

42  0 

S.  47  226 

8. 47  245 

9.  99  981 

0  18 

52  0 

8.  51  287 

8.51310 

9.99977 

0  8 

10 

S.  47  297 

8.47  316 

9.  99  981 

50 

10 

8.51351 

8.  51  374 

9.99977 

50 

20 

8.  47  368 

8.47  387 

9.99  981 

40 

20 

8.51416 

8.51439 

9.99977 

40 

30 

S.  47  439 

8.47  458 

9.  99  981 

30 

30 

8.  51  480 

8.  51  503 

9.99977 

30 

40 

8. 47  509 

8.47  528 

9.  99  981 

20 

40 

8.51544 

8.  51  568 

9.99  977 

20 

50 

S.  47  5S0 

8. 47  599 

9.  99  981 

10 

50 

8.  51  609 

8.  51  632 

9.99977 

10 

43  0 

8.47  650 

8.  47  669 

9.  99  981 

0  17 

53  0 

8.  51  673 

8.  51  696 

9.99977 

0  7 

10 

8.  47  720 

8. 47  740 

9.  99  980 

50 

10 

8.51737 

8.  51  760 

9.  99  976 

50 

20 

8. 47  790 

8.  47  810 

9.99980 

40 

20 

8.  51  801 

8.  51  824 

9.  99  976 

40 

30 

8.  47  860 

8. 47  880 

9.  99  980 

30 

30 

8.  51  864 

8.  51  8S8 

9.99976 

30 

40 

8.  47  930 

8.  47  950 

9.  99  980 

20 

40 

8.51928 

8.51952 

9.99976 

20 

50 

8.  48  000 

8. 48  020 

9. 99  980 

10 

50 

8.  51  992 

8.52  015 

9.  99  976 

10 

44  0 

8.  48  069 

8. 48  090 

9.  99  980 

0  16 

54  0 

8.52  055 

8.  52  079 

9. 99  976 

0   6 

10 

8.  4S  139 

8.  48  159 

9.  99  980 

50 

10 

8.  52  119 

8.  52  143 

9.99976 

50 

20 

8.  48  208 

8. 48  228 

9. 99  980 

40 

20 

8.  52  182 

8.  52  206 

9.99976 

40 

30 

8.  48  278 

8.  48  298 

9.  99  980 

30 

30 

8.  52  245 

8.52269 

9.99976 

30 

40 

8.  48  347 

8. 48  367 

9. 99  980 

20 

40 

8.  52  308 

8.  52  332 

9.99976 

20 

50 

S.  48  416 

8.  48  436 

9.  99  980 

10 

50 

8.52  371 

8.  52  396 

9.99976 

10 

45  0 

8.  48  485 

8.  48  505 

9.  99  980 

0  15 

55  0 

8.  52  434 

8.  52  459 

9.99  976 

0  5 

10 

8.48  554 

8.  48  574 

9.  99  980 

50 

10 

8.52  497 

8.  52  522 

9.  99  976 

50 

20 

8.  4S  622 

8.  48  643 

9.  99  980 

40 

20 

8.  52  560 

8.  52  584 

9.99976 

40 

30 

8.  48  691 

8.48  711 

9.  99  980 

30 

30 

8.  52  623 

8.  52  647 

9.99975 

30 

40 

8.  48  760 

8. 48  780 

9.  99  979 

20 

40 

8.  52  685 

8.  52  710 

9.99  975 

20 

50 

8. 48  828 

8. 48  849 

9.  99  979 

10 

50 

8.  52  748 

8.  52  772 

9.99  975 

10 

46  0 

8.  48  896 

8.48  917 

9.  99  979 

0  14 

56  0 

8.52  810 

8.  52  835 

9.  99  975 

0  4 

10 

8.  48  965 

8. 48  985 

9.  99  979 

50 

10 

8.  52  872 

8.  52  897 

9.  99  975 

50 

20 

8. 49  033 

8. 49  053 

9.  99  979 

40 

20 

8.  52  935 

8.  52  960 

9.99  975 

40 

30 

8.  49  101 

8.  49  121 

9.  99  979 

30 

30 

8.  52  997 

8.  53  022 

9. 99  975 

30 

40 

8. 49  169 

8.  49  189 

9.  99  979 

20 

40 

8.  53  059 

8.  53  084 

9. 99  975 

20 

50 

8. 49  236 

8.49  257 

9.  99  979 

10 

50 

8.  53  121 

8.  53  146 

9.99975 

10 

47  0 

8. 49  304 

8. 49  325 

9.  99  979 

0  13 

57  0 

8.  53  183 

8.  53  208 

9.99975 

0   3 

10 

8. 49  372 

8.  49  393 

9.  99  979 

50 

10 

8.  53  245 

8.  53  270 

9.99975 

50 

20 

8.  49  439 

8. 49  460 

9.  99  979 

40 

20 

8.  53  306 

8.  53  332 

9.99975 

40 

30 

8.  49  506 

8.  49  528 

9.  99  979 

30 

30 

8.  53  368 

8.  53  393 

9.  99  975 

30 

40 

8.  49  574 

8.  49  595 

9.  99  979 

20 

40 

8.  53  429 

8.  53  455 

9.  99  975 

20 

50 

8.  49  641 

8. 49  662 

9.  99  979 

10 

50 

8.  53  491 

8.53  516 

9.  99  974 

10 

48  0 

8. 49  708 

8.  49  729 

9.  99  979 

0  12 

58  0 

8.  53  552 

8.  53  578 

9.  99  974 

0   2 

10 

8. 49  775 

8. 49  796 

9.  99  979 

50 

10 

8.  53  614 

8.  53  639 

9.  99  974 

50 

20 

8.  49  842 

8.  49  863 

9.  99  978 

40 

20 

8.  53  675 

8. 53  700 

9.99974 

40 

30 

8.  49  908 

8. 49  930 

9.  99  978 

30 

30 

8.  53  736 

8.  53  762 

9.  99  974 

30 

40 

8. 49  975 

8. 49  997 

9.  99  978 

20 

40 

8.  53  797 

8.  53  823 

9.99974 

20 

50 

8.50  042 

8.  50  063 

9.  99  978 

10 

50 

8.  53  858 

8.  53  884 

9.  99  974 

10 

49  0 

8.  50  108 

8. 50  130 

9.99  978 

0   11 

59  0 

8.  53  919 

8.  53  945 

9.99974 

0    1 

10 

8.50174 

8.  50  196 

9. 99  978 

50 

10 

8.  53  979 

8.54  005 

9. 99  974 

50 

20 

8.  50  241 

8.  50  263 

9.  99  978 

40 

20 

8.  54  040 

8.  54  066 

9.  99  974 

40 

30 

8.50  307 

8.  50  329 

9. 99  978 

30 

30 

8.  54  101 

8.  54  127 

9.  99  974 

30 

40 

8.50  373 

8.  50  395 

9.  99  978 

20 

40 

8.  54  161 

8.  54  187 

9.  99  974 

20 

50 

8.  50  439 

8.  50  461 

9. 99  978 

10 

50 

8.  54  222 

8.  54  248 

9.99974 

10 

50  0 

8.  50  504 

8.  50  527 

9.  99  978 

0  10 

60  0 

8.  54  282 

8.  54  308 

9. 99  974 

0  0 

t  tt 

log  cos 

log  cot 

log  sin 

tt  t 

t  ft 

log  cos 

log  cot 

log  sin 

,,, 

88 


28 

1° 

_L 

log  sin 

8 

24  186 

log  tan 

8 

24  192 

log  cot 

•1  1 

log  cos 

g 

r 

0 

75  808 

99  993 

60 

1 

24  903 

24  910 

75  090 

99  993 

59 

2 

25  609 

25  616 

74  384 

99  993 

58 

3 

26  304 

26  312 

73  688 

99  993 

57 

4 

26  988 

26  996 

73  004 

99  992 

56 

5 

27  661 

27  669 

72  331 

99  992 

55 

6 

28  324 

28  332 

71668 

99  992 

54 

7 

28  977 

28  986 

71  014 

99  992 

53 

8 

29  621 

29  629 

70  371 

99  992 

52 

9 

30  255 

30  263 

69  737 

99  991 

51 

10 

30  879 

30  888 

69  112 

99  991 

50 

11 

31495 

31505 

68  495 

99  991 

49 

12 

32  103 

32  112 

67  888 

99  990 

48 

13 

32  702 

32  711 

67  289 

99  990 

47 

14 

33  292 

33  302 

66  698 

99  990 

46 

15 

33  875 

33  886 

66  114 

99  990 

45 

16 

34  450 

34  461 

65  539 

99  989 

44 

17 

35  018 

35  029 

64  971 

99  989 

43 

18 

35  578 

35  590 

64  410 

99  989 

42 

19 

36  131 

36  143 

63  857 

99  989 

41 

20 

36  678 

36  689 

63  311 

99  988 

40 

21 

37  217 

37  229 

62  771 

99  988 

39 

22 

37  750 

37  762 

62  238 

99  988 

38 

23 

38  276 

38  289 

61  711 

99  987 

37 

24 

38  796 

38  809 

61  191 

99  987 

36 

25 

39  310 

39  323 

60  677 

99  987 

35 

26 

39  818 

39  832 

60  168 

99  986 

34 

27 

40  320 

40  334 

59  666 

99  986 

33 

28 

40  816 

40  830 

59  170 

99  986 

32 

29 

41307 

41321 

58  679 

99  985 

31 

30 

41  792 

41807 

58  193 

99  985 

30 

31 

42  272 

42  287 

57  713 

99  985 

29 

32 

42  746 

42  762 

57  238 

99  984 

28 

33 

43  216 

43  232 

56  768 

99  984 

27 

34 

43  680 

43  696 

56  304 

99  984 

26 

35 

44  139 

44  156 

55  844 

99  983 

25 

36 

44  594 

44  611 

55  389 

99  983 

24 

37 

45  044 

45  061 

54  939 

99  983 

23 

38 

45  489 

45  507 

54  493 

99  982 

22 

39 

45  930 

45  948 

54  052 

99  982 

21 

40 

46  366 

46  385 

53  615 

99  982 

20 

41 

46  799 

46  817 

53  183 

99  981 

19 

42 

47  226 

47  245 

52  755 

99  981 

18 

43 

47  650 

47  669 

52  331 

99  981 

17 

44 

48  069 

48  089 

51911 

99  980 

16 

45 

48  485 

48  505 

51495 

99  980 

15 

46 

48  896 

48  917 

51083 

99  979 

14 

47 

49  304 

49  325 

50  675 

99  979 

13 

48 

49  70S 

49  729 

50  271 

99  979 

12 

49 

50  108 

50  130 

49  870 

99  978 

11 

50 

50  504 

50  527 

49  473 

99  978 

10 

51 

50  897 

50  920 

49  080 

99  977 

9 

52 

51  287 

51310 

48  690 

99  977 

8 

53 

51673 

51696 

48  304 

99  977 

7 

54 

52  055 

52  079 

47  921 

99  976 

6 

55 

52  434 

52  459 

47  541 

99  976 

5 

53 

52  810 

52  835 

47  165 

99  975 

4 

57 

53  183 

53  208 

46  792 

99  975 

3 

58 

53  552 

53  578 

46  422 

99  974 

2 

59 

53  919 

53  945 

46  055 

99  974 

1 

60 

54  2S2 
8 

54  308 

45  692 
H 

99  974 
g 

0 

f 

log  cos 

o 

log  cot 

log  tan 

log  sin 

f 

r 

log  sin 

o 

log  tan 

Q 

log  cot 

11 

log  cos 

n 

r 

0 

54  282 

o 

54  308 

45  692 

y 

99  974 

60 

1 

54  642 

54  669 

45  331 

99  973 

59 

2 

54  999 

55  027 

44  973 

99  973 

58 

3 

55  354 

55  382 

44  618 

99  972 

57 

4 

55  705 

55  734 

44  266 

99  972 

56 

5 

56  054 

56  083 

43  917 

99  971 

55 

6 

56  400 

56  429 

43  571 

99  971 

54 

7 

56  743 

56  773 

43  227 

99  970 

53 

8 

57  084 

57  114 

42  886 

99  970  % 

52 

9 

57  421 

57  452 

42  548 

99  969 

51 

10 

57  757 

57  788 

42  212 

99  969 

50 

11 

58  089 

58  121 

41879 

99  968 

49 

12 

58  419 

58  451 

41  549 

99  968 

48 

13 

58  747 

58  779 

41  221 

99  967 

47 

14 

59  072 

59  105 

40  895 

99  967 

46 

15 

59  395 

59  428 

40  572 

99  967 

45 

16 

59  715 

59  749 

40  251 

99  966 

44 

17 

60  033 

60  068 

39  932 

99  966 

43 

18 

60  349 

60  384 

39  616 

99  965 

42 

19 

60  662 

60  698 

39  302 

99  964 

41 

20 

60  973 

61009 

38  991 

99  964 

40 

21 

61  282 

61319 

38  681 

99  963 

39 

22 

61  589 

61  626 

38  374 

99  963 

38 

23 

61894 

61931 

38  069 

99  962 

37 

24 

62  196 

62  234 

37  766 

99  962 

36 

25 

62  497 

62  535 

37  465 

99  961 

35 

26 

62  795 

62  834 

37  166 

99  961 

34 

27 

63  091 

63  131 

36  869 

99  960 

33 

28 

63  385 

63  426 

36  574 

99  960 

32 

29 

63  678 

63  718 

36  282 

99  959 

31 

30 

63  968 

64  009 

35  991 

99  959 

30 

31 

64  256 

64  298 

35  702 

99  958 

29 

32 

64  543 

64  585 

35  415 

99  958 

28 

33 

64  827 

64  870 

35  130 

99  957 

27 

34 

65  110 

65  154 

34  846 

99  956 

26 

35 

65  391 

65  435 

34  565 

99  956 

25 

36 

65  670 

65  715 

34  285 

99  955 

24 

37 

65  947 

65  993 

34  007 

99  955 

23 

38 

66  223 

66  269 

33  731 

99  954 

22 

39 

66  497 

66  543 

33  457 

99  954 

21 

40 

66  769 

66  816 

33  184 

99  953 

20 

41 

67  039 

67  087 

32  913 

99  952 

19 

42 

67  308 

67  356 

32  644 

99  952 

18 

43 

67  575 

67  624 

32  376 

99  951 

17 

44 

67  841 

67  890 

32  110 

99  951 

16 

45 

68  104 

68  154 

31  846 

99  950 

15 

46 

68  367 

68  417 

31583 

99  949 

14 

47 

68  627 

68  678 

31  322 

99  949 

13 

48 

68  886 

68  938 

31062 

99  94S 

12 

49 

69  144 

69  196 

30  804 

99  94S 

11 

50 

69  400 

69  453 

30  547 

99  947 

10 

51 

69  654 

69  70S 

30  292 

99  946 

9 

52 

69  907 

69  962 

30  038 

99  946 

8 

53 

70  159 

70  214 

29  786 

99  945 

7 

54 

70  409 

70  465 

29  535 

99  944 

6 

55 

70  658 

70  714 

29  286 

99  944 

5 

56 

70  905 

70  962 

29  038 

99  943 

4 

57 

71  151 

71  208 

2S792 

99  942 

3 

58 

71  395 

71453 

28  547 

99  942 

2 

59 

71638 

71  697 

28  303 

99  941 

1 

60 

71880 

o 

71940 

o, 

28  060 
H 

99  940 

9 

log  sin 

0 

r 

f 

0 

log  cos 

o 

log  cot 

log  tan 

88 


87 


39 


; 

log  sin 

q 

log  tan 

8 

71  940 

log  cot 

1  1 

log  cos 

Q 

f 

0 

718SO 

28  060 

y 
99  940 

60 

1 

72  120 

72  181 

27  819 

99  940 

59 

2 

72  359 

72  420 

27  580 

99  939 

58 

3 

72  597 

72  659 

27  341 

99  938 

57 

4 

72  834 

72  896 

27  104 

99  938 

56 

5 

73  069 

73  132 

26  868 

99  937 

55 

6 

73  303 

73  366 

26  634 

99  936 

54 

7 

73  535 

73  600 

26  400 

99  936 

53 

8 

73  767 

73  832 

26  168 

99  935 

52 

9 

73  997 

74  063 

25  937 

99  934 

51 

10 

74  226 

74  292 

25  708 

99  934 

50 

11 

74  454 

74  521 

25  479 

99  933 

49 

12 

74  680 

74  748 

25  252 

99  932 

48 

13 

74  906 

74  974 

25  026 

99  932 

47 

14 

75  130 

75  199 

24  801 

99  931 

46 

15 

75  353 

75  423 

24  577 

99  930 

45 

16 

75  575 

75  645 

24  355 

99  929 

44 

17 

75  795 

75  867 

24  133 

99  929 

43 

18 

76  015 

76  087 

23  913 

99  928 

42 

19 

76  234 

76  306 

23  694 

99  927 

41 

20 

76  451 

76  525 

23  475 

99  926 

40 

21 

76  667 

76  742 

23  258 

99  926 

39 

22 

76  883 

76  958 

23  042 

99  925 

38 

23 

77  097 

V  173 

22  827 

99  924 

37 

24 

77  310 

77  387 

22  613 

99  923 

36 

25 

77  522 

77  600 

22  400 

99  923 

35 

26 

77  733 

77  811 

22  189 

99  922 

34 

27 

77  943 

78  022 

21978 

99  921 

33 

28 

7S152 

78  232 

21  768 

99  920 

32 

29 

78  360 

78  441 

21559 

99  920 

31 

30 

78  568 

78  649 

21351 

99  919 

30 

31 

78  774 

78  855 

21  145 

99  918 

29 

32 

78  979 

79  061 

20  939 

99  917 

28 

33 

79  183 

79  266 

20  734 

99  917 

27 

34 

79  386 

79  470 

20  530 

99  916 

26 

35 

79  588 

79  673 

20  327 

99  915 

25 

36 

79  789 

79  875 

20  125. 

99  914 

24 

37 

79  990 

80  076 

19  924 

99  913 

23 

38 

80  189 

80  277 

19  723 

99  913 

22 

39 

80  388 

80  476 

19  524 

99  912 

21 

40 

80  585 

80  674 

19  326 

99  911 

20 

41 

80  782 

80  872 

19  128 

99  910 

19 

42 

80  978 

81068 

18  932 

99  909 

18 

43 

81  173 

81  264 

18  736 

99  909 

17 

44 

81  367 

81459 

18  541 

99  908 

16 

45 

81  560 

81  653 

18  347 

99  907 

15 

46 

81  752 

81846 

18  154 

99  906 

14 

47 

81944 

82  038 

17  962 

99  905 

13 

48 

82  134 

82  230 

17  770 

99  904 

12 

49 

82  324 

82  420 

17  580 

99  904 

11 

50 

82  513 

82  610 

17  390 

99  903 

10 

51 

82  701 

82  799 

17  201 

99  902 

9 

52 

82  888 

82  987 

17  013 

99  901 

8 

53 

83  07£ 

83  175 

16  825 

99  900 

7 

54 

83  261 

83  361 

16  639 

99  899 

6 

55 

83  446 

83  547 

16  453 

99  898 

5 

56 

83  630 

83  732 

16  268 

99  898 

4 

57 

83  813 

83  916 

16  084 

99  897 

3 

58 

83  996 

84  100 

15  900 

99  896 

2 

59 

84  177 

84  282 

15  718 

99  895 

1 

60 

84  358 

8 

84  464 

o 

15  536 
n 

99  894 
o 

0 

t 

o 

log  COS 

o 

log  cot 

— xi — 
log  tan 

a 

log  sin 

r 

4° 

29 

t 

log  sin 

log  tan 

8 

84  464 

log  cot 
—ll- 
lS  536 

log  cos 

n 

? 

0 

84  358 

y 
99S94 

60 

1 

84  539 

84  646 

15  354 

99  893 

59 

2 

84  718 

84  826 

15  174 

99  892 

58 

3 

84  897 

85  006 

14  994 

99  891 

57 

4 

85  075 

85  185 

14  815 

99  891 

56 

5 

85  252 

85  363 

14  637 

99  890 

55 

6 

85  429 

85  540 

14  460 

99  889 

54 

7 

85  605 

85  717 

14  283 

99  888 

53 

8 

85  780 

85  893 

14  107 

99  887 

52 

9 

85  955 

86  069 

13  931 

99  886 

51 

10 

86  128 

86  243 

13  757 

99  885 

50 

11 

86  301 

86  417 

13  583 

99  884 

49 

12 

86  474 

86  591 

13  409 

99  883 

48 

13 

86  645 

86  763 

13  237 

99  882 

47 

14 

86  816 

86  935 

13  065 

99  881 

46 

15 

86  987 

87  106 

12  894 

99  880 

45 

16 

87  156 

87  277 

12  723 

99  879 

44 

17 

87  325 

87  447 

12  553 

99  879 

43 

18 

87  494 

87  616 

12  3S4 

99  878 

42 

19 

87  661 

87  785 

12  215 

99  877 

41 

20 

87  829 

87  953 

12  047 

99  876 

40 

21 

87  995 

88  120 

11  880 

99  875 

39 

22 

88  161 

88  287 

11  713 

99  874 

38 

23 

88  326 

88  453 

11547 

99  873 

37 

24 

88  490 

88  618 

11382 

99  872 

36 

25 

88  654 

88  783 

11  217 

99  871 

35 

26 

88  817 

88  948 

11052 

99  870 

34 

27 

88  980 

89  111 

10  889 

99  869 

33 

28 

89  142 

89  274 

10  726 

99  868 

32 

29 

89  304 

89  437 

10  563 

99  867 

31 

30 

89  464 

89  598 

10  402 

99S66 

30 

31 

89  625 

89  760 

10  240 

99  865 

29 

32 

89  784 

89  920 

10  080 

99  864 

28 

33 

89  943 

90  080 

09  920 

99  863 

27 

34 

90  102 

90  240 

09  760 

99  862 

26 

35 

90  260 

90  399 

09  601 

99  861 

25 

36 

90  417 

90  557 

09  443 

99  860 

24 

37 

90  574 

90  715 

09  285 

99  859 

23 

38 

90  730 

90  872 

09  128 

99  858 

22 

39 

90  885 

91029 

08  971 

99  857 

21 

40 

91040 

91  185 

08  815 

99  856 

20 

41 

91  195 

91340 

08  660 

99  855 

19 

42 

91349 

91495 

08  505 

99  854 

18 

43 

91  502 

91650 

08  350 

99  853 

17 

44 

91655 

91803 

08  197 

99  852 

16 

45 

91807 

91957 

08  043 

99  851 

15 

46 

91959 

92  110 

07  890 

99  850 

14 

47 

92  110 

92  262 

07  738 

99  848 

13 

48 

92  261 

92  414 

07  586 

99  847 

12 

49 

92  411 

92  565 

07  435 

99  846 

11 

50 

92  561 

92  716 

07  284 

99  845 

10 

51 

92  710 

92  866 

07  134 

99  844 

9 

52 

92  859 

93  016 

06  984 

99  843 

8 

53 

93  007 

93  165 

06  835 

99  842 

7 

54 

93  154 

93  313 

06  687 

99  841 

6 

55 

93  301 

93  462 

06  538 

99  840 

5 

56 

93  448 

93  609 

06  391 

99  839 

4 

57 

93  594 

93  756 

06  244 

99  838 

3 

58 

93  740 

93  903 

06  097 

99  837 

2 

59 

93  885 

94  049 

05  951 

99  836 

1 

60 

94  030 

o 

94  195 
— 8 — 

log  cot 

05  805 
n 

99  834 

0 

t 

o 

log  cos 

j.  j. 
log  tan 

y 

log  sin 

r 

86 


85 


30 

5° 

r 

log  sin 

log  tan 

log  cot 

log  cos 

t 

8 

8 

— 11 — 

9 

0 

94  030 

94  195 

05  805 

99  834 

60 

1 

94  174 

94  340 

05  660 

99  833 

59 

2 

94  317 

94  485 

05  515 

99  832 

58 

3 

94  461 

94  630 

05  370 

99  831 

57 

4 

94  603 

94  773 

05  227 

99  830 

56 

5 

94  746 

94  917 

05  083 

99  829 

55 

6 

94  887 

95  060 

04  940 

99  828 

54 

7 

95  029 

95  202 

04  798 

99  827 

53 

8 

95  170 

95  344 

04  656 

99  825 

52 

9 

95  310 

95  486 

04  514 

99  824 

51 

10 

95  450 

95  627 

04  373 

99  823 

50 

11 

95  589 

95  767 

04  233 

99  822 

49 

12 

95  72S 

95  908 

04  092 

99  821 

48 

13 

95  867 

96  047 

03  953 

99  820 

47 

14 

96  005 

96  187 

03  813 

99  819 

46 

15 

96  143 

96  325 

03  675 

99  817 

45 

16 

96  280 

96  464 

03  536 

99  816 

44 

17 

96  417 

96  602 

03  398 

99  815 

43 

18 

96  553 

96  739 

03  261 

99  814 

42 

19 

96  689 

96  877 

03  123 

99  813 

41 

20 

96  825 

97  013 

02  987 

99  812 

40 

21 

96  960 

97  150 

02  850 

99  810 

39 

22 

97  095 

97  285 

02  715 

99  809 

38 

23 

97  229 

97  421 

02  579 

99  808 

37 

24 

97  363 

97  556 

02  444 

99  807 

36 

25 

97  496 

97  691 

02  309 

99  806 

35 

26 

97  629 

97  825 

02  175 

99  804 

34 

27 

97  762 

97  959 

02  041 

99  803 

33 

28 

97  894 

98  092 

01908 

99  802 

32 

29 

98  026 

98  225 

01  775 

99  801 

31 

30 

98  157 

98  358 

01642 

99  800 

30 

31 

98  288 

98  490 

01  510 

99  798 

29 

32 

98  419 

98  622 

01378 

99  797 

28 

33 

98  549 

98  753 

01247 

99  796 

27 

34 

98  679 

98  884 

01  116 

99  795 

26 

35 

98  808 

99  015 

00  985 

99  793 

25 

36 

98  937 

99  145 

00  855 

99  792 

24 

37 

99  066 

99  275 

00  725 

99  791 

23 

38 

99  194 

99  405 

00  595 

99  790 

22 

39 

99  322 

99  534 

00  466 

99  788 

21 

40 

99  450 

99  662 

00  338 

99  787 

20 

41 

99  577 

99  791 

00  209 

99  786 

19 

42 
43 

99  704 
99  830 

18?  213 

90  081 

99  785 
99  783 

18 
17 

66644 

9$  954 

44 
45 

99  956 

00  174 
00  301 

99  826 
99  699 

99  782 
99  781 

16 
15 

00  082 

46 

00  207 

00  427 

99  573 

99  780 

14 

47 

00  332 

00  553 

99  447 

99  778 

13 

48 

00  456 

00  679 

99  321 

99  777 

12 

49 

00  581 

00  805 

99  195 

99  776 

11 

50 

00  704 

00  930 

99  070 

99  775 

10 

51 

00  828 

01055 

98  945 

99  773 

9 

52 

00  951 

01  179 

98  821 

99  772 

8 

53 

01074 

01303 

98  697 

99  771 

7 

54 

01  196 

01427 

98  573 

99  769 

6 

55 

01318 

01  550 

98  450 

99  768 

5 

56 

01440 

01673 

98  327 

99  767 

4 

57 

01561 

01  796 

98  204 

99  765 

3 

58 

01682 

01918 

98  082 

99  764 

2 

59 

01803 

02  040 

97  960 

99  763 

1 

60 

01923 

Q 

02  162 

97  838 
in 

99  761 
o 

• 

f 

a 

log  COS 

9 

log  cot 

1U 

log  tan 

y 

log  sin 

■ 

6 


f 

log  sin 

Q 

log  tan 

9 

02  162 

log  cot 
— 10— 

97  838 

log  cos 

f 

0 

01923 

99  761 

60 

1 

02  043 

02  283 

97  717 

99  760 

59 

2 

02  163 

02  404 

97  596 

99  759 

58 

3 

02  283 

02  525 

97  475 

99  757 

57 

4 

02  402 

02  645 

97  355 

99  756 

56 

5 

02  520 

02  766 

97  234 

99  755 

55 

6 

02  639 

02  885 

97  115 

99  753 

54 

7 

02  757 

03  005 

96  995 

99  752 

53 

8 

02  874 

03  124 

96  876 

99  751 

52 

9 

02  992 

03  242 

96  758 

99  749 

51 

10 

03  109 

03  361 

96  639 

99  748 

50 

11 

03  226 

03  479 

96  521 

99  747 

49 

12 

03  342 

03  597 

96  403 

99  745 

48 

13 

03  458 

03  714 

96  286 

99  744 

47 

14 

03  574 

03  832 

96  168 

99  742 

46 

15 

03  690 

03  948 

96  052 

99  741 

45 

16 

03  805 

04  065 

95  935 

99  740 

44 

17 

03  920 

04  181 

95  819 

99  738 

43 

18 

04  034 

04  297 

95  703 

99  737 

42 

19 

04  149 

04  413 

95  587 

99  736 

41 

20 

04  262 

04  528 

95  472 

99  734 

40 

21 

04  376 

04  643 

95  357 

99  733 

39 

22 

04  490 

04  758 

95  242 

99  731 

38 

23 

04  603 

04  873 

95  127 

99  730 

37 

24 

04  715 

04  987 

95  013 

99  728 

36 

25 

04  828 

05  101 

94  899 

99  727 

35 

26 

04  940 

05  214 

94  786 

99  726 

34 

27 

05  052 

05  328 

94  672 

99  724 

33 

28 

05  164 

05  441 

94  559 

99  723 

32 

29 

05  275 

05  553 

94  447 

99  721 

31 

30 

05  386 

05  666 

94  334 

99  720 

30 

31 

05  497 

05  778 

94  222 

99  718 

29 

32 

05  607 

05  890 

94  110 

99  717 

28 

33 

05  717 

06  002 

93  998 

99  716 

27 

34 

05  827 

06  113 

93  887 

99  714 

26 

35 

05  937 

06  224 

93  776 

99  713 

25 

36 

06  046 

06  335 

93  665 

99  711 

24 

37 

06  155 

06  445 

93  555 

99  710 

23 

38 

06  264 

06  556 

93  444 

99  708 

22 

39 

06  372 

06  666 

93  334 

99  707 

21 

40 

06  481 

06  775 

93  22£ 

99  705 

20 

41 

06  589 

06  885 

93  115 

99  704 

19 

42 

06  696 

06  994 

93  006 

99  702 

18 

43 

06  804 

07  103 

92  897 

99  701 

17 

44 

06  911 

07  211 

92  789 

99  699 

16 

45 

07  018 

07  320 

92  680 

99  698 

15 

46 

07  124 

07  428 

92  572 

99  696 

14 

47 

07  231 

07  536 

92  464 

99  695 

13 

48 

07  337 

07  643 

92  357 

99  693 

12 

49 

07  442 

07  751 

92  249 

99  692 

11 

50 

07  548 

07  858 

92  142 

99  690 

10 

51 

07  653 

07  964 

92  036 

99  689 

9 

52 

07  758 

08  071 

91  929 

99  687 

8 

53 

07  863 

08  177 

91  823 

99  686 

7 

54 

07  968 

08  283 

91  717 

99  684 

6 

55 

08  072 

08  389 

91611 

99  683 

5 

56 

08  176 

08  495 

91505 

99  681 

4 

57 

08  280 

08  600 

91400 

99  6S0 

3 

58 

08  383 

08  705 

91295 

99  678 

2 

59 

08  486 

08  810 

91  190 

99  677 

1 

|60 

OS  589 
o 

08  914 

Q 

91086 
in 

99  675 

n 

0 

|  ' 

y 

log  cos 

y 

log  cot 

— J.U — 

log  tan 

y 
log  sin 

f 

84' 


83 


f 

log  sin 

n 

log  tan 

Q 

log  cot 

log  cos 
n 

r 

0 

08  589 

08  914 

1U 

91086 

99  675 

60 

1 

08  692 

09  019 

90  981 

99  674 

59 

2 

08  795 

09  123 

90  877 

99  672 

58 

3 

08  897 

09  227 

90  773 

99  670 

57 

4 

08  999 

09  330 

90  670 

99  669 

56 

5 

09  101 

09  434 

90  566 

99  667 

55 

6 

09  202 

09  537 

90  463 

99  666 

54 

7 

09  304 

09  640 

90  360 

99  664 

53 

8 

09  405 

09  742 

90  258 

99  663 

52 

9 

09  506 

09  845 

90  155 

99  661 

51 

10 

09  606 

09  947 

90  053 

99  659 

50 

11 

09  707 

10  049 

89  951 

99  658 

49 

12 

09  807 

10  150 

89  850 

99  656 

48 

13 

09  907 

10  252 

89  748 

99  655 

47 

14 

10  006 

10  353 

89  647 

99  653 

46 

15 

10  106 

10  454 

89  546 

99  651 

45 

16 

10  205 

10  555 

89  445 

99  650 

44 

17 

10  304 

10  656 

89  344 

99  648 

43 

18 

10  402 

10  756 

89  244 

99  647 

42 

19 

10  501 

10  856 

89  144 

99  645 

41 

20 

10  599 

10  956 

89  044 

99  643 

40 

21 

10  697 

11056 

88  944 

99  642 

39 

22 

10  795 

11  155 

88  845 

99  640 

38 

23 

10  893 

11  254 

88  746 

99  638 

37 

24 

10  990 

11353 

88  647 

99  637 

36 

25 

11087 

11452 

88  548 

99  635 

35 

26 

11  184 

11551 

88  449 

99  633 

34 

27 

11281 

11649 

88  351 

99  632 

33 

28 

11377 

11  747 

88  253 

99  630 

32 

29 

11474 

11845 

88  155 

99  629 

31 

30 

11570 

11943 

88  057 

99  627 

30 

31 

11666 

12  040 

87  960 

99  625 

29 

32 

11  761 

12  138 

87  862 

99  624 

28 

33 

11857 

12  235 

87  765 

99  622 

27 

34 

11952 

12  332 

87  668 

99  620 

26 

35 

12  047 

12  428 

87  572 

99  618 

25 

36 

12  142 

12  525 

87  475 

99  617 

24 

37 

12  236 

12  621 

S7  379 

99  615 

23 

38 

12  331 

12  717 

87  283 

99  613 

22 

39 

12  425 

12  813 

87  187 

99  612 

21 

40 

12  519 

12  909 

87  091 

99  610 

20 

41 

12  612- 

13  004 

86  996 

99  608 

19 

42 

12  706 

13  099 

86  901 

99  607 

18 

43 

12  799 

13  194 

86  806 

99  605 

17 

44 

12  892 

13  289 

86  711 

99  603 

16 

45 

12  985 

13  384 

86  616 

99  601 

15 

46 

13  078 

13  478 

86  522 

99  600 

14 

47 

13  171 

13  573 

86  427 

99  598 

13 

48 

13  263 

13  667 

86  333 

99  596 

12 

49 

13  355 

13  761 

86  239 

99  595 

11 

50 

13  447 

13  854 

86  146 

99  593 

10 

51 

13  539 

13  948 

86  052 

99  591 

9 

52 

13  630 

14  041 

85  959 

99  589 

8 

53 

13  722 

14  134 

85  866 

99  588 

7 

54 

13  813 

14  227 

85  773 

99  586 

6 

55 

13  904 

14  320 

85  680 

99  584 

5 

56 

13  994 

14  412 

85  588 

99  582 

4 

57 

14  085 

14  504 

85  496 

99  581 

3 

58 

14  175 

14  597 

85  403 

99  579 

2 

59 

14  266 

14  688 

85  312 

99  577 

1 

60 

14  356 

Q 

14  780 

— 9 — 

log  cot 

85  220 
in 

99  575 

n 

0 

f 

a 

log  COS 

1U 

log  tan 

y 

log  sin 

r 

8° 

31 

r 

log  sin 

Q 

log  tan 

Q 

log  cot 
in 

log  cos 

g 

t 

0 

14  356 

14  780 

— ±u — 

85  220 

99  575 

60 

1 

14  445 

14  872 

85  128 

99  574 

59 

2 

14  535 

14  963 

85  037 

99  572 

58 

3 

14  624 

15  054 

84  946 

99  570 

57 

4 

14  714 

15  145 

84  855 

99  568 

56 

5 

14  803 

15  236 

84  764 

99  566 

55 

6 

14  891 

15  327 

84  673 

99  565 

54 

7 

14  980 

15  417 

84  583 

99  563 

53 

8 

15  069 

15  508 

84  492 

99  561 

52 

9 

15  157 

15  598 

84  402 

99  559 

51 

10 

15  245 

15  688 

84  312 

99  557 

50 

11 

15  333 

15  777 

84  223 

99  556 

49 

12 

15  421 

15  867 

84  133 

99  554 

48 

13 

15  508 

15  956 

84  044 

99  552 

47 

14 

15  596 

16  046 

83  954 

99  550 

46 

15 

15  683 

16  135 

83  865 

99  548 

45 

16 

15  770 

16  224 

83  776 

99  546 

44 

17 

15  857 

16  312 

83  688 

99  545 

43 

18 

15  944 

16  401 

83  599 

99  543 

42 

19 

16  030 

16  489 

83  511 

99  541 

41 

20 

16  116 

16  577 

83  423 

99  539 

40 

21 

16  203 

16  665 

83  335 

99  537 

39 

22 

16  289 

16  753 

83  247 

99  535 

38 

23 

16  374 

16  841 

83  159 

99  533 

37 

24 

16  460 

16  928 

83  072 

99  532 

36 

25 

16  545 

17  016 

82  984 

99  530 

35 

26 

16  631 

17  103 

82  897 

99  528 

34 

27 

16  716 

17  190 

82  810 

99  526 

33 

28 

16  801 

17  277 

82  723 

99  524 

32 

29 

16  886 

17  363 

82  637 

99  522 

31 

30 

16  970 

17  450 

82  550 

99  520 

30 

31 

17  055 

17  536 

82  464 

99  518 

29 

32 

17  139 

17  622 

82  378 

99  517 

28 

33 

17  223 

17  708 

82  292 

99  515 

27 

34 

17  307 

17  794 

82  206 

99  513 

26 

35 

17  391 

17  880 

82  120 

99  511 

25 

36 

17  474 

17  965 

82  035 

99  509 

24 

37 

17  558 

18  051 

81949 

99  507 

23 

38 

17  641 

18  136 

81864 

99  505 

22 

39 

17  724 

18  221 

81  779 

99  503 

21 

40 

17  807 

18  306 

81694 

99  501 

20 

41 

17  890 

18  391 

81609 

99  499 

19 

42 

17  973 

18  475 

81525 

99  497 

18 

43 

18  055 

18  560 

81440 

99  495 

17 

44 

18  137 

18  644 

81356 

99  494 

16 

45 

18  220 

18  728 

81272 

99  492 

15 

46 

18  302 

18  812 

81  188 

99  490 

14 

47 

18  383 

18  896 

81  104 

99  488 

13 

48 

18  465 

18  979 

81021 

99  486 

12 

49 

18  547 

19  063 

80  937 

99  484 

11 

50 

18  628 

19  146 

80  854 

99  482 

10 

51 

18  709 

19  229 

80  771 

99  480 

9 

52 

18  790 

19  312 

80  688 

99  478 

8 

53 

18  871 

19  395 

80  605 

99  476 

7 

54 

18  952 

19  478 

80  522 

99  474 

6 

55 

19  033 

19  561 

80  439 

99  472 

5 

56 

19  113 

19  643 

80  357 

99  470 

4 

57 

19  193 

19  725 

80  275 

99  468 

3 

58 

19  273 

19  807 

80  193 

99  466 

2 

59 

19  353 

19  889 

80  111 

99  464 

1 

60 

19  433 

Q 

19  971 
— 9 — 

log  cot 

80  029 
—10— 

log  tan 

99  462 
9 

0 

r 

y 

log  cos 

log  sin 

r 

82( 


81 


32 

9° 

0 

log  sin 
9 

19  433 

log  tan 

9  — 

19  971 

log  cot 

20 

log  cos 
g 

f 

80  029 

99  462 

60 

1 

19  513 

20  053 

79  947 

99  460 

59 

2 

19  592 

20  134 

79  866 

99  458 

58 

3 

19  672 

20  216 

79  784 

99  456 

57 

4 

19  751 

20  297 

79  703 

99  454 

56 

5 

19  830 

20  378 

79  622 

99  452 

55 

6 

19  909 

20  459 

79  541 

99  450 

54 

7 

19  988 

20  540 

79  460 

99  448 

53 

8 

20  067 

20  621 

79  379 

99  446 

52 

9 

20  145 

20  701 

79  299 

99  444 

51 

10 

20  223 

20  782 

79  218 

99  442 

50 

11 

20  302 

20  862 

79  138 

99  440 

49 

12 

20  380 

20  942 

79  058 

99  438 

48 

13 

20  458 

21  022 

78  978 

99  436 

47 

14 

20  535 

21  102 

78  898 

99  434 

46 

15 

20  613 

21  182 

78  818 

99  432 

45 

16 

20  691 

21  261 

78  739 

99  429 

44 

17 

20  768 

21341 

78  659 

99  427 

43 

18 

20  845 

21  420 

78  580 

99  425 

42 

19 

20  922 

21499 

78  501 

99  423 

41 

20 

20  999 

21  578 

78  422 

99  421 

40 

21 

21  076 

21657 

78  343 

99  419 

39 

22 

21  153 

21  736 

78  264 

99  417 

38 

23 

21  229 

21814 

78  186 

99  415 

37 

24 

21306 

21893 

78  107 

99  413 

36 

25 

21382 

21971 

78  029 

99  411 

35 

26 

2145S 

22  049 

77  951 

99  409 

34 

27 

21534 

22  127 

77  873 

99  407 

33 

28 

21  610 

22  205 

77  795 

99  404 

32 

29 

21685 

22  283 

77  717 

99  402 

31 

30 

21  761 

22  361 

77  639 

99  400 

30 

31 

21  836 

22  438 

77  562 

99  398 

29 

32 

21912 

22  516 

77  484 

99  396 

28 

33 

21987 

22  593 

77  407 

99  394 

27 

34 

22  062 

22  670 

77  330 

99  392 

26 

35 

22  137 

22  747 

77  253 

99  390 

25 

36 

22  211 

22  824 

77  176 

99  388 

24 

37 

22  286 

22  901 

77  099 

99  385 

23 

38 

22  361 

22  977 

77  023 

99  383 

22 

39 

22  435 

23  054 

76  946 

99  3S1 

21 

40 

22  509 

23  130 

76  870 

99  379 

20 

41 

22  583 

23  206 

76  794 

99  377 

19 

42 

22  657 

23  283 

76  717 

99  375 

18 

43 

22  731 

23  359 

76  641 

99  372 

17 

44 

22  805 

23  435 

76  565 

99  370 

16 

45 

22  878 

23  510 

76  490 

99  368 

15 

46 

22  952 

23  5S6 

76  414 

99  366 

14 

47 

23  025 

23  661 

76  339 

99  364 

13 

48 

23  098 

23  737 

76  263 

99  362 

12 

49 

23  171 

23  812 

76  188 

99  359 

11 

50 

23  244 

23  887 

76  113 

99  357 

10 

51 

23  317 

23  962 

76  038 

99  355 

9 

52 

23  390 

24  037 

75  963 

99  353 

8 

53 

23  462 

24  112 

75  888 

99  351 

7 

54 

23  535 

24  186 

75  814 

99  348 

6 

55 

23  607 

24  261 

75  739 

99  346 

5 

56 

23  679 

24  335 

75  665 

99  344 

4 

57 

23  752 

24  410 

75  590 

99  342 

3 

58 

23  823 

24  484 

75  516 

99  340 

2 

59 

23  895 

24  558 

75  442 

99  337 

1 

60 

23  967 

Q 

24  632 

Q 

75  368 
in 

99  335 

g 

0 

f 

y 

log  cos 

log  cot 

— 1U — 

log  tan 

log  sin 

t 

10 


t 

log  sin 

log  tan 

log  cot 
in 

log  cos 

9 

99  335 

t 
60 

0 

23  967 

y 
24  632 

75  368 

1 

24  039 

24  706 

75  294 

99  333 

59 

2 

24  110 

24  779 

75  221 

99  331 

58 

3 

24  181 

24  853 

75  147 

99  328 

57 

4 

24  253 

24  926 

75  074 

99  326 

56 

5 

24  324 

25  000 

75  000 

99  324 

55 

6 

24  395 

25  073 

74  927 

99  322 

54 

7 

24  466 

25  146 

74  854 

99  319 

53 

8 

24  536 

25  219 

74  781 

99  317 

52 

9 

24  607 

25  292 

74  708 

99  315 

51 

10 

24  677 

25  365 

74  635 

99  313 

50 

11 

24  748 

25  437 

74  563 

99  310 

49 

12 

24  818 

25  510 

74  490 

99  308 

48 

13 

24  888 

25  582 

74  418 

99  306 

47 

14 

24  958 

25  655 

74  345 

99  304 

46 

15 

25  028 

25  727 

74  273 

99  301 

45 

16 

25  098 

25  799 

74  201 

99  299 

44 

17 

25  168 

25  871 

74  129 

99  297 

43 

18 

25  237 

25  943 

74  057 

99  294 

42 

19 

25  307 

26  015 

73  985 

99  292 

41 

20 

25  376 

26  086 

73  914 

99  290 

40 

21 

25  445 

26  158 

73  842 

99  288 

39 

22 

25  514 

26  229 

73  771 

99  285 

38 

23 

25  583 

26  301 

73  699 

99  283 

37 

24 

25  652 

26  372 

73  628 

99  281 

36 

25 

25  721 

26  443 

73  557 

99  278 

35 

26 

25  790 

26  514 

73  486 

99  276 

34 

27 

25  858 

26  585 

73  415 

99  274 

33 

28 

25  927 

26  655 

73  345 

99  271 

32 

29 

25  995 

26  726 

73  274 

99  269 

31 

30 

26  063 

26  797 

73  203 

99  267 

30 

31 

26  131 

26  867 

73  133 

99  264 

29 

32 

26  199 

26  937 

73  063 

99  262 

28 

33 

26  267 

27  008 

72  992 

99  260 

27 

34 

26  335 

27  078 

72  922 

99  257 

26 

35 

26  403 

27  14S 

72  852 

99  255 

25 

36 

26  470 

27  218 

72  7S2 

99  252 

24 

37 

26  538 

27  288 

72  712 

99  250 

23 

38 

26  605 

27  357 

72  643 

99  24S 

22 

39 

26  672 

27  427 

72  573 

99  245 

21 

40 

26  739 

27  496 

72  504 

99  243 

20 

41 

26  806 

27  566 

72  434 

99  241 

19 

42 

26  873 

27  635 

72  365 

99  238 

18 

43 

26  940 

27  704 

72  296 

99  236 

17 

44 

27  007 

27  773 

72  227 

99  233 

16 

45 

27  073 

27  842 

72  158 

99  231 

15 

46 

27  140 

27  911 

72  0S9 

99  229 

14 

47 

27  206 

27  9S0 

72  020 

99  226 

13 

48 

27  273 

28  049 

71  951 

99  224 

12 

49 

27  339 

28  117 

71  883 

99  221 

11 

50 

27  405 

28  186 

71814 

99  219 

10 

51 

27  471 

28  254 

71  746 

99  217 

9 

52 

27  537 

28  323 

71677 

99  214 

8 

53 

27  602 

28  391 

71  609 

99  212 

7 

54 

27  668 

28  459 

71  541 

99  209 

6 

55 

27  734 

28  527 

71  473 

99  207 

5 

56 

27  799 

28  595 

71  405 

99  204 

4 

57 

27  864 

28  662 

71338 

99  202 

a 

58 

27  930 

28  730 

71  270 

99  200 

2 

59 

27  995 

2S  798 

71  202 

99  197 

1 

60 

2S060 
o 

28  865 

71  135 
10 

99  195 
q 

0 

f 

y 

log  003 

y 

log  cot 

log  tan 

log  sin 

' 

80( 


79 


lr 


r 

log  sin 
o 

log  tan 

Q 

log  cot 
in 

log  cos 

9 

99  195 

t 
60 

0 

y 

28  060 

2SS65 

1U 

71  135 

1 

28  125 

28  933 

71067 

99  192 

59 

2 

28  190 

29  000 

71000 

99  190 

58 

3 

2S  254 

29  067 

70  933 

99  187 

57 

4 

28  319 

29  134 

70  866 

99  185 

56 

5 

28  384 

29  201 

70  799 

99  182 

55 

6 

2S44S 

29  268 

70  732 

99  180 

54 

7 

2S512 

29  335 

70  665 

99  177 

53 

8 

28  577 

29  402 

70  598 

99  175 

52 

9 

28  641 

29  46S 

70  532 

99  172 

51 

10 

28  705 

29  535 

70  465 

99  170 

50 

11 

28  769 

29  601 

70  399 

99  167 

49 

12 

28S33 

29  668 

70  332 

99  165 

48 

13 

28  S96 

29  734 

70  266 

99  162 

47 

14 

2S960 

29  800 

70  200 

99  160 

46 

15 

29  024 

29  866 

70  134 

99  157 

45 

16 

29  0S7 

29  932 

70  068 

99  i55 

44 

17 

29  150 

29  998 

70  002 

99  152 

43 

18 

29  214 

30  064 

69  936 

99  150 

42 

19 

29  277 

30  130 

69S70 

99  147 

41 

20 

29  340 

30  195 

69  805 

99  145 

40 

21 

29  403 

30  261 

69  739 

99  142 

39 

22 

29  466 

30  326 

69  674 

99  140 

38 

23 

29  529 

30  391 

69  609 

99  137 

37 

24 

29  591 

30  457 

69  543 

99  135 

36 

25 

29  654 

30  522 

69  478 

99  132 

35 

26 

29  716 

30  587 

69  413 

99  130 

34 

27 

29  779 

30  652 

69  34S 

99  127 

33 

28 

29  841 

30  717 

69  283 

99  124 

32 

29 

29  903 

30  782 

69  218 

99  122 

31 

.30 

29  966 

30  846 

69  154 

99  119 

30 

31 

30  028 

30  911 

69  0S9 

99  117 

29 

32 

30  090 

30  975 

69  025 

99  114 

28 

33 

30  151 

31040 

68  960 

99  112 

27 

34 

30  213 

31  104 

68  896 

99  109 

26 

35 

30  275 

31  168 

68  832 

99  106 

25 

36 

30  336 

31  233 

68  767 

99  104 

24 

37 

30  398 

31297 

68  703 

99  101 

23 

38 

30  459 

31361 

68  639 

99  099 

22 

39 

30  521 

31425 

68  575 

99  096 

21 

40 

30  582 

31489 

68  511 

99  093 

20 

41 

30  643 

31  552 

68  448 

99  091 

19 

42 

30  704 

31616 

68  384 

99  088 

18 

43 

30  765 

31  679 

68  321 

99  086 

17 

44 

30  826 

31743 

68  257 

99  0S3 

16 

45 

30  887 

31806 

68  194 

99  080 

15 

46 

30  947 

31870 

68  130 

99  078 

14 

47 

31008 

31933 

68  067 

99  075 

13  ' 

48 

31068 

31996 

68  004 

99  072 

12 

49 

31  129 

32  059 

67  941 

99  070 

11 

50 

31  189 

32  122 

67  878 

99  067 

10 

51 

31  250 

32  185 

67  815 

99  064 

9 

52 

31310 

32  248 

67  752 

99  062 

8 

53 

31370 

32  311 

67  689 

99  059 

7 

54 

31430 

32  373 

67  627 

99  056 

6 

55 

31490 

32  436 

67  564 

99  054 

5 

56 

31549 

32  498 

67  502 

99  051 

4 

57 

31  609 

32  561 

67  439 

99  048 

3 

58 

31669 

32  623 

67  377 

99  046 

2 

59 

31  728 

32  685 

67  315 

99  043 

1 

60 

31  788 

32  747 

67  253 

99  040 

0 

o 

9 

log  cot 

—10  — 

log  tan 

o 

r 

o 

log  COS 

y 
log  sin 

r 

12° 

33 

r 

log  sin 

9 

31  788 

log  tan 
9 

32  747 

log  cot 
— 10— 
67  253 

log  cos 

9 

99  040 

f 

0 

60 

1 

31847 

32  810 

67  190 

99  038 

59 

2 

31907 

32  872 

67  128 

99  035 

58 

3 

31  966 

32  933 

67  067 

99  032 

57 

4 

32  025 

32  995 

67  005 

99  030 

56 

5 

32  084 

33  057 

66  943 

99  027 

55 

6 

32  143 

33  119 

66  881 

99  024 

54 

7 

32  202 

33  180 

66  820 

99  022 

53 

8 

32  261 

33  242 

66  758 

99  019 

52 

9 

32  319 

33  303 

66  697 

99  016 

51 

10 

32  378 

33  365 

66  635 

99  013 

50 

11 

32  437 

33  426 

66  574 

99  011 

49 

12 

32  495 

33  487 

66  513 

99  008 

48 

13 

32  553 

33  548 

66  452 

99  005 

47 

14 

32  612 

33  609 

66  391 

99  002 

46 

15 

32  670 

33  670 

66  330 

99  000 

45 

16 

32  728 

33  731 

66  269 

98  997 

44 

17 

32  786 

33  792 

66  208 

98  994 

43 

18 

32  844 

33  853 

66  147 

98  991 

42 

19 

32  9Q2 

33  913 

66  087 

98  989 

41 

20 

32  960 

33  974 

66  026 

98  986 

40 

21 

33  01S 

34  034 

65  966 

98  983 

39 

22 

33  075 

34  095 

65  905 

9*8  980 

38  1 

23 

33  133 

34  155 

65  845 

98  978 

37 

24 

33  190 

34  215 

65  785 

98  975 

36 

25 

33  248 

34  276 

65  724 

98  972 

35 

26 

33  305 

34  336 

65  664 

98  969 

34 

27 

33  362 

34  396 

65  604 

98  967 

33 

28 

33  420 

34  456 

65  544 

98  964 

32 

29 

33  477 

34  516 

65  484 

98  961 

31 

30 

33  534 

34  576 

65  424 

98  958 

30 

31 

33  591 

34  635 

65  365 

98  955 

29 

32 

33  647 

34  695 

65  305 

98  953 

28 

33 

33  704 

34  755 

65  245 

98  950 

27 

34 

33  761 

34  814 

65  186 

98  947 

26 

35 

33  818 

34  874 

65  126 

98  944 

25 

36 

33  874 

34  933 

65  067 

98  941 

24 

37 

33  931 

34  992 

65  008 

98  938 

23 

38 

33  987 

35  051 

64  949 

98  936 

22 

39 

34  043 

35  111 

64  889 

98  933 

21 

40 

34  100 

35  170 

64  830 

98  930 

20 

41 

34  156 

35  229 

64  771 

98  927 

19 

42 

34  212 

35  288 

64  712 

98  924 

18 

43 

34  268 

35  347 

64  653 

98  921 

17 

44 

34  324 

35  405 

64  595 

98  919 

16 

45 

34  380 

35  464 

64  536 

98  916 

15 

46 

34  436 

35  523 

64  477 

98  913 

14 

47 

34  491 

35  581 

64  419 

98  910 

13 

48- 

34  547 

35  640 

64  360 

98  907 

12 

49 

34  602 

35  698 

64  302 

98  904 

11 

50 

34  658 

35  757 

64  243 

98  901 

10 

51 

34  713 

35  815 

64  185 

98  898 

9 

52 

34  769 

35  873 

64  127 

98  896 

8 

53 

34  824 

35  931 

64  069 

98  893 

7 

54 

34  879 

35  989 

64  011 

98  890 

6 

55 

34  934 

36  047 

63  953 

98  887 

5 

56 

34  989 

36  105 

63  895 

98  884 

4 

57 

35  044 

36  163 

63  837 

98  881 

3 

58 

35  099 

36  221 

63  779 

98  878 

2 

59 

35  154 

36  279 

63  721 

98  875 

1 

60 

35  209 

9 — 

log  cos 

36  336 

9 

log  cot 

63  664 
—  10— 

log  tan 

98  872 
o 

0 

r 

y 

log  sin 

f 

78 


77' 


34 

13° 

r 

log  sin 

O 

log  tan 

log  cot 
in 

log  cos 
a 

r 

0 

35  209 

y 
36  336 

xu 

63  664 

98  872 

60 

1 

35  263 

36  394 

63  606 

98  869 

59 

2 

35  318 

36  452 

63  548 

98  867 

58 

3 

35  373 

36  509 

63  491 

98  864 

57 

4 

35  427 

36  566 

63  434 

98  861 

56 

5 

35  481 

36  624 

63  376 

98  858 

55 

6 

35  536 

36  681 

63  319 

98  855 

54 

7 

35  590 

36  738 

63  262 

98  852 

53 

8 

35  644 

36  795 

63  205 

98  849 

52 

9 

35  698 

36  852 

63  148 

98  846 

51 

10 

35  752 

36  909 

63  091 

98  843 

50 

11 

35  806 

36  966 

63  034 

98  840 

49 

12 

35  860 

37  023 

62  977 

98  837 

48 

13 

35  914 

37  080 

62  920 

98  834 

47 

14 

35  968 

37  137 

62  863 

98  831 

46 

15 

36  022 

37  193 

62  807 

98  828 

45 

16 

36  075 

37  250 

62  750 

98  825 

44 

17 

36  129 

37  306 

62  694 

98  822 

43 

18 

36  182 

37  363 

62  637 

98  819 

42 

19 

36  236 

37  419 

62  581 

98  816 

41 

20 

36  289 

37  476 

62  524 

98  813 

40 

21 

36  342 

37  532 

62  468 

98  810 

39 

22 

36  395* 

37  588 

62  412 

98  807 

38 

23 

36  449 

37  644 

62  356 

98  804 

37 

24 

36  502 

37  700 

62  300 

98  801 

36 

25 

36  555 

37  756 

62  244 

98  798 

35 

26 

36  608 

37  812 

62  188 

98  795 

34 

27 

36  660 

37  868 

62  132 

98  792 

33 

28 

36  713 

37  924 

62  076 

98  789 

32 

29 

36  766 

37  980 

62  020 

98  786 

31 

30 

36  819 

38  035 

61965 

98  783 

30 

31 

36  871 

38  091 

61909 

98  780 

29 

32 

36  924 

38  147 

61  853 

98  777 

28 

33 

36  976 

38  202 

61  798 

98  774 

27 

34 

37  028 

38  257 

61  743 

98  771 

26 

35 

37  0S1 

38  313 

61687 

98  768 

25 

36 

37  133 

38  368 

61632 

98  765 

24 

37 

37  185 

38  423 

61577 

98  762 

23 

38 

37  237 

38  479 

61521 

98  759 

22 

39 

37  289 

38  534 

61466 

98  756 

21 

40 

37  341 

38  589 

61411 

98  753 

20 

41 

37  393 

38  644 

61356 

98  750 

19 

42 

37  445 

38  699 

61  301 

98  746 

18 

43 

37  497 

38  754 

61  246 

98  743 

17 

44 

37  549 

38  808 

61  192 

98  740 

16 

45 

37  600 

38  863 

61  137 

98  737 

15 

46 

37  652 

38  918 

61  082 

98  734 

14 

47 

37  703 

38  972 

61  028 

98  731 

13 

48 

37  755 

39  027 

60  973 

98  728 

12 

49 

37  806 

39  082 

60  918 

98  725 

11 

50 

37  858 

39  136 

60  864 

98  722 

10 

51 

37  909 

39  190 

60  810 

98  719 

9 

52 

37  960 

39  245 

60  755 

98  715 

8 

53 

38  011 

39  299 

60  701 

98  712 

7 

54 

38  062 

39  353 

60  647 

98  709 

6 

55 

38  113 

39  407 

60  593 

98  706 

5 

56 

38  164 

39  461 

60  539 

98  703 

4 

57 

38  215 

39  515 

60  485 

98  700 

3 

58 

38  266 

39  569 

60  431 

98  697 

2 

59 

38  317 

39  623 

60  377 

98  694 

1 

60 

38  368 
o 

39  677 

■ 

60  323 
in 

98  690 
o 

0 

T 

u 

log  COS 

y 

log  cot 

1U 

log  tan 

y 

log  sin 

f 

14( 


t 

log  sin 
g 

log  tan 

Q 

log  cot 

^Q 

log  cos 

Q 

t 

0 

38  368 

39  677 

60  323 

98  690 

60 

1 

38  418 

39  731 

60  269 

98  687 

59 

2 

38  469 

39  785 

60  215 

98  684 

58 

3 

38  519 

39  838 

60  162 

9S681 

57 

4 

38  570 

39  892 

60  108 

98  678 

56 

5 

38  620 

39  945 

60  055 

98  675 

55 

6 

38  670 

39  999 

60  001 

98  671 

54 

7 

38  721 

40  052 

59  948 

98  668 

53 

8 

38  771 

40  106 

59  894 

98  665 

52 

9 

38  821 

40  159 

59  841 

98  662 

51 

10 

38  871 

40  212 

59  788 

98  659 

50 

11 

38  921 

40  266 

59  734 

98  656 

49 

12 

38  971 

40  319 

59  681 

98  652 

48 

13 

39  021 

40  372 

59  628 

98  649 

47 

14 

39  071 

40  425 

59  575 

98  646 

46 

15 

39  121 

40  478 

59  522 

98  643 

45 

16 

39  170 

40  531 

59  469 

98  640 

44 

17 

39  220 

40  584 

59  416 

98  636 

43 

18 

39  270 

40  636 

59  364 

98  633 

42 

19 

39  319 

40  689 

59  311 

98  630 

41 

20 

39  369 

40  742 

59  258 

98  627 

40 

21 

39  418 

40  795 

59  205 

98  623 

39 

22 

39  467 

40  847 

59  153 

98  620 

38 

23 

39  517 

40  900 

59  100 

98  617 

37 

24 

39  566 

40  952 

59  048 

98  614 

36 

25 

39  615 

41005 

58  995 

98  610 

35 

26 

39  664 

41057 

58  943 

98  607 

34 

27 

39  713 

41  109 

58  891 

98  604 

33 

28 

39  762 

41  161 

58  839 

98  601 

32 

29 

39  811 

41  214 

58  786 

98  597 

31 

30 

39  860 

41266 

58  734 

98  594 

30 

31 

39  909 

41318 

58  682 

98  591 

29 

32 

39  958 

41  370 

58  630 

98  5S8 

28 

33 

40  006 

41422 

58  578 

9S5S4 

27 

34 

40  055 

41474 

58  526 

98  581 

26 

35 

40  103 

41  526 

58  474 

9S  57S 

25 

36 

40  152 

41578 

58  422 

9S574 

24 

37 

40  200 

41629 

58  371 

9S571 

23 

38 

40  249 

41681 

58  319 

98  568 

22 

39 

40  297 

41  733 

58  267 

98  565 

21 

40 

40  346 

41  784 

SS  216 

98  561 

20 

41 

40  394 

41836 

58  164 

9S  55S 

19 

42 

40  442 

418S7 

58  113 

98  555 

18 

43 

40  490 

41  939 

58  061 

98  551 

17 

44 

40  538 

41990 

58  010 

98  548 

16 

45 

40  586 

42  041 

57  959 

98  545 

15 

46 

40  634 

42  093 

57  907 

98  541 

14 

47 

40  682 

42  144 

57  856 

98  538 

13 

48 

40  730 

42  195 

57  805 

98  535 

12 

49 

40  778 

42  246 

57  754 

QS  531 

11 

50 

40  825 

42  297 

57  703 

9S  528 

10 

51 

40  873 

42  348 

57  652 

98  525 

9 

52 

40  921 

42  399 

57  601 

98  521 

8 

53 

40  968 

42  450 

57  550 

9S  5 IS 

7 

54 

41016 

42  501 

57  499 

98  515 

6 

55 

41063 

42  552 

57  448 

98  511 

5 

56 

41  111 

42  603 

57  397 

98  508 

4 

57 

41  158 

42  653 

57  347 

98  505 

3 

58 

41  205 

42  704 

57  296 

98  501 

2 

59 

41  252 

42  755. 

57  245 

98  498 

1 

60 

41300 

42  805 

Q 

57  195. 
in 

98  494 

0 

r 

y 

log  cos 

y 

log  cot 

log  tan 

y 

log  sin 

t 

76' 


75' 


15 


f 

log  sin 
o 

log  tan 

A 

log  cot 
—10— 
57  195 

log  cos 

A 

t 

0 

41300 

42  805 

98  494 

60 

1 

41347 

42  856 

57  144 

98  491 

59 

2 

41394 

42  906 

57  094 

98  488 

58 

3 

41441 

42  957 

57  043 

98  484 

57 

4 

41488 

43  007 

56  993 

98  481 

56 

5 

41535. 

43  057 

56  943 

98  477 

55 

6 

41  582 

43  108 

56  892 

98  474 

54 

7 

41628 

43  158 

56  842 

98  471 

53 

8 

41675 

43  208 

56  792 

98  467 

52 

9 

41  722 

43  258 

56  742 

98  464 

51 

10 

41  768 

43  308 

56  692 

98  460 

50 

11 

41815 

43  358 

56  642 

98  457 

49 

12 

41861 

43  408 

56  592 

98  453 

48 

13 

41908 

43  458 

56  542 

98  450 

47 

14 

41954 

43  508 

56  492 

98  447 

46 

15 

42  001 

43  558 

56  442 

98  443 

45 

16 

42  047 

43  607 

56  393 

98  440 

44 

17 

42  093 

43  657 

56  343 

98  436 

43 

18 

42  140 

43  707 

56  293 

98  433 

42 

19 

42  186 

43  756 

56  244 

98  429 

41 

20 

42  232 

43  806 

56  194 

98  426 

40 

21 

42  278 

43  855 

56  145 

98  422 

39 

22 

42  324 

43  905 

56  095 

98  419 

38 

23 

42  370 

43  954 

56  046 

98  415 

37 

24 

42  416 

44  004 

55  996 

98  412 

36 

25 

42  461 

44  053 

55  947 

98  409 

35 

26 

42  507 

44  102 

55  898 

98  405 

34 

27 

42  553 

44  151 

55  849 

98  402 

33 

28 

42  599 

44  201 

55  799 

98  398 

32 

29 

42  644 

44  250 

55  750 

98  395 

31 

30 

42  690 

44  299 

55  701 

98  391 

30 

31 

42  735 

44  348 

55  652 

98  388 

29 

32 

42  781 

44  397 

55  603 

98  384 

28 

33 

42  826 

44  446 

55  554 

98  381 

27. 

34 

42  872 

44  49£ 

55  505 

98  377 

26 

35 

42  917 

44  544 

55  456 

98  373 

25 

36 

42  962 

44  592 

55  408 

98  370 

24 

37 

43  008 

44  641 

55  359 

98  366 

23 

38 

43  053 

44  690 

55  310 

98  363 

22 

39 

43  098 

44  738 

55  262 

98  359 

21 

40 

43  143 

44  787 

55  213 

98  356 

20 

41 

43  188 

44  836 

55  164 

98  352 

19 

42 

43  233 

44  884 

55  116 

98  349 

18 

43 

43  278 

44  933 

55  067 

98  345 

17 

44 

43  323 

44  981 

55  019 

98  342 

16 

45 

43  367 

45  029 

54  971 

98  338 

15 

46 

43  412 

45  078 

54  922 

98  334 

14 

47 

43  457 

45  126 

54  874 

98  331 

13 

48 

43  502 

45  174 

54  826 

98  327 

12 

49 

43  546 

45  222 

54  778 

98  324 

11 

50 

43  591 

45  271 

54  729 

98  320 

10 

51 

43  635 

45  319 

54  681 

98  317 

9 

52 

43  680 

45  367 

54  633 

98  313 

8 

53 

43  724 

45  415 

54  585 

98  309 

7 

54 

43  769 

45  463 

54  537 

98  306 

6 

55 

43  813 

45  511 

54  489 

98  302 

5 

56 

43  857 

45  559 

54  441 

98  299 

4 

57 

43  901 

45  606 

54  394 

98  295 

3 

58 

43  946 

45  654- 

54  346 

98  291 

2 

59 

43  990 

45  702 

54  298 

98  288 

1 

60 

44  034 

Q 

45  750 

A 

54  250 
in 

98  284 

0 

f 

a 

log  COS 

log  cot 

1U 

log  tan 

y 
log  sin 

r 

16° 

35 

f 

log  sin 

log  tan 

A 

log  cot 

1  ft 

log  cos 
n 

? 

0 

44  034 

45  750 

— ±u — 

54  250 

98  284 

60 

1 

44  078 

45  797 

54  203 

98  281 

59 

2 

44  122 

45  845 

54  155 

98  277 

58 

3 

44  166 

45  892 

54  108 

98  273 

57 

4 

44  210 

45  940 

54  060 

98  270 

56 

5 

44  253 

45  987 

54  013 

98  266 

55 

6 

44  297 

46  035 

53  965 

98  262 

54 

7 

44  341 

46  082 

53  918 

98  259 

53 

8 

44  385 

46  130 

53  870 

Q8  255 

52 

9 

44  428 

46  177 

53  823 

98  251 

51 

10 

44  472 

46  224 

53  776 

98  248 

50 

11 

44  516 

46  271 

53  729 

98  244 

49 

12 

44  559 

46  319 

53  681 

98  240 

48 

13 

44  602 

46  366 

53  634 

98  237 

47 

14 

44  646 

46  413 

53  587 

98  233 

46 

15 

44  689 

46  460 

53  540 

98  229 

45 

16 

44  733 

46  507 

53  493 

98  226 

44 

17 

44  776 

46  554 

53  446 

98  222 

43 

18 

44  819 

46  601 

53  399 

98  218 

42 

19 

44  862 

46  648 

53  352 

98  215 

41 

20 

44  905 

46  694 

53  306 

98  211 

40 

21 

44  948 

46  741 

53  259 

98  207 

39 

22 

44  992 

46  788 

53  212 

98  204 

38 

23 

45  035 

46  835 

53  165 

98  200 

37 

24 

45  077 

46  881 

53  119 

98  196 

36 

25 

45  120 

46  928 

53  072 

98  192 

35 

26 

45  163 

46  975 

53  025 

98  189 

34 

27 

45  206 

47  021 

52  979 

98  185 

33 

28 

45  249 

47  068 

52  932 

98  181 

32 

29 

45  292 

47  114 

52  886 

98  177 

31 

30 

45  334 

47  160 

52  840 

98  174 

30 

31 

45  377 

47  207 

52  793 

98  170 

29 

32 

45  419 

47  253 

52  747 

98  166 

28 

33 

45  462 

47  299 

52  701 

98  162 

27 

34 

45  504 

47  346 

52  654 

98  159 

26 

35 

45  547 

47  392 

52  608 

98  155 

25 

36 

45  589 

47  438 

52  562 

98  151 

24 

37 

45  632 

47  484 

52  516 

98  147 

23 

38 

45  674 

47  530 

52  470 

98  144 

22 

39 

45  716 

47  576 

52  424 

98  140 

21 

40 

45  758 

47  622 

52  378 

98  136 

20 

41 

45  801 

47  668 

52  332 

98  132 

19 

42 

45  843 

47  714 

52  286 

98  129 

18 

43 

45  885 

47  760 

52  240 

98  125 

17 

44 

45  927 

47  806 

52  194 

98  121 

16 

45 

45  969 

47  852 

52  148 

98  117 

15 

46 

46  011 

47  897 

52  103 

98  113 

14 

47 

46  053 

47  943 

52  057 

98  110 

13 

48 

46  095 

47  989 

52  011 

98  106 

12 

49 

46  136 

48  035 

51965 

98  102 

11 

50 

46  178 

48  080 

51920 

98  098 

10 

51 

46  220 

48  126 

51874 

98  094 

9 

52 

46  262 

48  171 

51829 

98  090 

8 

53 

46  303 

48  217 

51  783 

98  087 

7 

54 

46  345 

48  262 

51  738 

98  083 

6 

55 

46  386 

48  307 

51693 

98  079 

5 

56 

46  428 

48  353 

51647 

98  075 

4 

57 

46  469 

48  398 

51  602 

98  071 

3 

58 

46  511 

48  443 

51557 

98  067 

2 

59 

46  552 

48  489 

51511 

98  063 

1 

60 

46  594 

—9 

log  cos 

48  534 

— 9 

log  cot 

51466 
— 10  — 

98  060 
g 

0 

f 

log  tan 

log  sin 

f 

74( 


73 


36 

17° 

18° 

r 

log  sin 

n 

log  tan 

Q 

log  cot 

-\Q 

log  cos 

9 

98  060 

r 

r 

log  sin 

Q 

log  tan 
g 

log  cot 
Jft 

log  C08 
n 

t 

0 

46  594 

48  534 

51466 

60 

0 

y 

48  998 

51  178 

48  822 

97  821 

60 

1 

46  635 

48  579 

51421 

98  056 

59 

1 

49  037 

51  221 

48  779 

97  817 

59 

2 

46  676 

48  624 

51376 

98  052 

58 

2 

49  076 

51264 

48  736 

97  812 

58 

3 

46  717 

48  669 

51331 

98  048 

57 

3 

49  115 

51306 

48  694 

97  808 

57 

4 

46  758 

48  714 

51  286 

98  044 

56 

4 

49  153 

51349 

48  651 

97  804 

56 

5 

46  800 

48  759 

51241 

98  040 

55 

5 

49  192 

51392 

48  608 

97  800 

55 

6 

46  841 

48  804 

51  196 

98  036 

54 

6 

49  231 

51435 

48  565 

97  796 

54 

7 

46  882 

48  849 

51  151 

98  032 

53 

7 

49  269 

51478 

48  522 

97  792 

53 

8 

46  923 

48  894 

51  106 

98  029 

52 

8 

49  308 

51  520 

48  480 

97  788 

52 

9 

46  964 

48  939 

51061 

98  025 

51 

9 

49  347 

51563 

48  437 

97  784 

51 

10 

47  005 

48  984 

51016 

98  021 

50 

10 

49  385 

51606 

48  394 

97  779 

50 

11 

47  045 

49  029 

50  971 

98  017 

49 

11 

49  424 

51648 

48  352 

97  775 

49 

12 

47  086 

49  073 

50  927 

98  013 

48 

12 

49  462 

51691 

48  309 

97  771 

48 

13 

47  127 

49  118 

50  882 

98  009 

47 

13 

49  500 

51  734 

48  266 

97  767 

47 

14 

47  168 

49  163 

50  837 

98  005 

46 

14 

49  539 

51776 

48  224 

97  763 

46 

15 

47  209 

49  207 

50  793 

98  001 

45 

15 

49  577 

51819 

48  181 

97  759 

45 

16 

47  249 

49  252 

50  748 

97  997 

44 

16 

49  615 

51861 

48  139 

97  754 

44 

17 

47  290 

49  296 

50  704 

97  993 

43 

17 

49  654 

51903 

48  097 

97  750 

43 

18 

47  330 

49  341 

50  659 

97  989 

42 

18 

49  692 

51946 

48  054 

97  746 

42 

19 

47  371 

49  385 

50  615 

97  986 

41 

19 

49  730 

51988 

48  012 

97  742 

41 

20 

47  411 

49  430 

50  570 

97  982 

40 

20 

49  768 

52  031 

47  969 

97  738 

40 

21 

47  452 

49  474 

50  526 

97  978 

39 

21 

49  806 

52  073 

47  927 

97  734 

39 

22 

47  492 

49  519 

50  481 

97  974 

38 

22 

49  844 

52  115 

47  885 

97  729 

38 

23 

47  533 

49  563 

50  437 

97  970 

37 

23 

49  882 

52  157 

47  843 

97  725 

37 

24 

47  573 

49  607 

50  393 

97  966 

36 

24 

49  920 

52  200 

47  800 

97  721 

36 

25 

47  613 

49  652 

50  348 

97  962 

35 

25 

49  958 

52  242 

47  758 

97  717 

35 

26 

47  654 

49  696 

50  304 

97  958 

34 

26 

49  996 

52  284 

47  716 

97  713 

34 

27 

47  694 

49  740 

50  260 

97  954 

33 

27 

50  034 

52  326 

47  674 

97  708 

33 

28 

47  734 

49  784 

50  216 

97  950 

32 

28 

50  072 

52  368 

47  632 

97  704 

32 

29 

47  774 

49  828 

50  172 

97  946 

31 

29 

50  110 

52  410 

47  590 

97  700 

31 

30 

47  814 

49  872 

50  128 

97  942 

30 

30 

50  148 

52  452 

47  548 

97  696 

30 

31 

47  854 

49  916 

50  084 

97  938 

29 

31 

50  185 

52  494 

47  506 

97  691 

29 

32 

47  894 

49  960 

50  040 

97  934 

28 

32 

50  223 

52  536 

47  464 

97  687 

28 

33 

47  934 

50  004 

49  996 

97  930 

27 

33 

50  261 

52  578 

47  422 

97  683 

27 

34 

47  974 

50  048 

49  952 

97  926 

26 

34 

50  298 

52  620 

47  380 

97  679 

26 

35 

48  014 

50  092 . 

49  908 

97  922 

25 

35 

50  336 

52  661 

47  339 

97  674 

25 

36 

48  054 

50  136 

49  864 

97  918 

24 

36 

50  374 

52  703 

47  297 

97  670 

24 

37 

48  094 

50  180 

49  820 

97  914 

23 

37 

50  411 

52  745 

47  255 

97  666 

23 

38 

48  133 

50  223 

49  777 

97  910 

22 

38 

50  449 

52  787 

47  213 

97  662 

22 

39 

48  173 

50  267 

49  733 

97  906 

21 

39 

50  486 

52  829 

47  171 

97  657 

21 

40 

48  213 

50  311 

49  689 

97  902 

20 

40 

50  523 

52  870 

47  130 

97  653 

20 

41 

48  252 

50  355 

49  645 

97  898 

19 

41 

50  561 

52  912 

47  088 

97  649 

19 

42 

48  292 

50  398 

49  602 

97  894 

18 

42 

50  598 

52  953 

47  047 

97  645 

18 

43 

48  332 

50  442 

49  558 

97  890 

17 

43 

50  635 

52  995 

47  005 

97  640 

17 

44 

4S371 

50  485 

49  515 

97  886 

16 

44 

50  673 

53  037 

46  963 

97  636 

16 

45 

48  411 

50  529 

49  471 

97  882 

15 

45 

50  710 

53  078 

46  922 

97  632 

15 

46 

48  450 

50  572 

49  428 

97  878 

14 

46 

50  747 

53  120 

46  880 

97  628 

14 

47 

48  490 

50  616 

49  384 

97  874 

13 

47 

50  784 

53  161 

46  839 

97  623 

13 

48 

48  529 

50  659 

49  341 

97  870 

12 

48 

50  821 

53  202 

46  79S 

97  619 

12 

49 

48  568 

50  703 

49  297 

97  866 

11 

49 

50  858 

53  244 

46  756 

97  615 

11 

50 

48  607 

50  746 

49  254 

97  861 

10 

50 

50  896 

53  285 

46  715 

97  610 

10 

51 

48  647 

50  789 

49  211 

97  857 

9 

51 

50  933 

53  327 

46  673 

97  606 

9 

52 

48  686 

50  833 

49  167 

97  853 

8 

52 

50  970 

53  368 

46  632 

97  602 

8 

53 

48  725 

50  876 

49  124 

97  849 

7 

53 

51007 

53  409 

46  591 

97  597 

7 

54 

48  764 

50  919 

49  081 

97  845 

6 

54 

51043 

53  450 

46  550 

97  593 

6 

55 

48  803 

50  962 

49  038 

97  841 

5 

55 

51080 

53  492 

46  508 

97  589 

5 

56 

48  842 

51005 

48  995 

97  837 

4 

56 

51  117 

53  533 

46  467 

97  584 

4 

57 

48  881 

51048 

48  952 

97  833 

3 

57 

51  154 

53  574 

46  426 

97  5S0 

3 

58 

48  920 

51  092 

48  908 

97  829 

2 

58 

51  191 

53  615 

46  385 

97  576 

2 

59 

48  959 

51  135 

48  865 

97  825 

1 

59 

51227 

53  656 

46  344 

97  571 

1 

60 

48  998 

9 

log  cos 

51  178 

9 — 

log  cot 

48  822 
—10  — 

log  tan 

97  821 
n 

0 

J60 

51264 

Q 

53  697 

n 

46  303 
in 

97  567 

Q 

0 

r 

y 

log  sin 

' 

1  < 

log  cos   log  cot 

log  tan 

y 

log  sin 

f 

72c 


71 


19 


r 

log  sin 
g 

log  tan 

9 

53  697 

log  cot 

10 

log  cos 

r 

0 

51264 

46  303 

97  567 

60 

1 

51301 

53  738 

46  262 

97  563 

59 

2 

51338 

53  779 

46  221 

97  558 

58 

3 

51374 

53  820 

46  180 

97  554 

57 

4 

51411 

53  861 

46  139 

97  55.5 

56 

5 

51  4+7 

53  902 

46  098 

97  545 

55 

6 

51484 

53  943 

46  057 

97  541 

54 

7 

51520 

53  984 

46  016 

97  536 

53 

8 

51557 

54  025 

45  975 

97  532 

52 

9 

51  593 

54  065 

45  935 

97  528 

51 

10 

51  629 

54  106 

45  894 

97  523 

50 

11 

51666 

54  147 

45  853 

97  519 

49 

12 

51  702 

54  187 

45  813 

97  515 

48 

13 

51  738 

54  228 

45  772 

97  510 

47 

14 

51  774 

54  269 

45  731 

97  506 

46- 

15 

51811 

54  309 

45  691 

97  501 

45 

16 

51  847 

54  350 

45  650 

97  497 

44 

17 

51883 

54  390 

45  610 

97  492 

43 

18 

51  919 

54  431 

45  569 

97  488 

42 

19 

51955 

54  471 

45  529 

97  484 

41 

20 

51991 

54  512 

45  488 

97  479 

40 

21 

52  027 

54  552 

45  448 

97  475 

39 

22 

52  063 

54  593 

45  407 

97  470 

38 

23 

52  099 

54  633 

45  367 

97  466 

37 

24 

52  135 

54  673 

45  327 

97  461 

36 

25 

52  171 

54  714 

45  286 

97  457 

35 

26 

52  207 

54  754 

45  246 

97  453 

34 

27 

52  242 

54  794 

45  206 

97  448 

33 

28 

52  278 

54  835 

45  165 

97  444 

32 

29 

52  314 

54  875 

45  125 

97  439 

31 

30 

52  350 

54  915 

45  085 

97  435 

30 

31 

52  385 

54  955 

45  045 

97  430 

29 

32 

52  421 

54  995 

45  005 

97  426 

28 

33 

52  456 

55  035 

44  965 

97  421 

27 

34 

52  492 

55  075 

44  925 

97  417 

26 

35 

52  527 

55  115 

44  885 

97  412 

25 

36 

52  563 

55  155 

44  845 

97  408 

24 

37 

52  598 

55  195 

44  805 

97  403 

23 

38 

52  634 

55  235 

44  765 

97  399 

22 

39 

52  669 

55  275 

44  725 

97  394 

21 

40 

52  705 

55  315 

44  685 

97  390 

20 

41 

52  740 

55  355 

44  645 

97  385 

19 

42 

52  775 

55  395 

44  605 

97  381 

18 

43 

52  811 

55  434 

44  566 

97  376 

17 

44 

52  846 

55  474 

44  526 

97  372 

16 

45 

52  881 

55  514 

44  486 

97  367 
97  363 

15 

46 

52  916 

55  554 

44  446 

14 

47 

52  951 

55  593 

44  407 

97  358 

13 

48 

52  986 

55  633 

44  367 

97  353 

12 

49 

53  021 

55  673 

44  327 

97  349 

11 

50 

53  056 

55  712 

44  288 

97  344 

10 

51 

53  092 

55  752 

44  248 

97  340 

9 

52 

53  126 

55  791 

44  209 

97  335 

8 

53 

53  161 

55  831 

44  169 

97  331 

7 

54 

53  196 

55  870 

44  130 

97  326 

6 

55 

53  231 

55  910 

44  090 

97  322 

5 

56 

53  266 

55  949 

44  051 

97  317 

4 

57 

53  301 

55  989 

44  011 

97  312 

3 

58 

53  336 

56  028 

43  972 

97  308 

2 

59 

53  370 

56  067 

43  933 

97  303 

1 

160 

53  405 
n 

56  107 

g_« 

43  893 

in 

97  299 

o 

0 

J  ' 

o 

log  COS 

log  cot 

— lu 

log  tan 

j 

log  sin 

/ 

20° 

37 

r 

log  sin 

log  tan 

g 

log  cot 

1Q 

log  cos 

9 

97  299 

t 

60 

0 

53  405 

56  107 

43  893 

1 

53  440 

56  146 

43  854 

97  294 

59 

2 

53  475 

56  185 

43  815 

97  289 

58 

3 

53  509 

56  224 

43  776 

97  285 

57 

4 

53  544 

56  264 

43  736 

97  280 

56 

5 

53  578 

56  303 

43  697 

97  276 

55 

6 

53  613 

56  342 

43  658 

97  271 

54 

7 

53  647 

56  381 

43  619 

97  266 

53 

8 

53  682 

56  420 

43  580 

97  262 

52 

9 

53  716 

56  459 

43  541 

97  257 

51 

10 

53  751 

56  498 

43  502 

97  252 

50 

11 

53  785 

56  537 

43  463 

97  248 

49 

12 

53  819 

56  576 

43  424 

97  243 

48 

13 

53  854 

56  615 

43  385 

97  238 

47 

14 

53  888 

56  654 

43  346 

97  234 

46 

15 

53  922 

56  693 

43  307 

97  229 

45 

16 

53  957 

56  732 

43  268 

97  224 

44 

17 

53  991 

56  771 

43  229 

97  220 

43 

18 

54  025 

56  810 

43  190 

97  215 

42 

19 

54  059 

56  849 

43  151 

97  210 

41 

20 

54  093 

56  887 

43  113 

97  206 

40 

21 

54  127 

56  926 

43  074 

97  201 

39 

22 

54  161 

56  965 

43  035 

97  196 

38 

23 

54  195 

57  004 

42  996 

97  192 

37 

24 

54  229 

57  042 

42  958 

97  187 

36 

25 

54  263 

57  081 

42  919 

97  182 

35 

26 

54  297 

57  120 

42  880 

97  178 

34 

27 

54  331 

57  158 

42  842 

97  173 

33 

28 

54  365 

57  197 

42  803 

97  168 

32 

29 

54  399 

57  235 

42  765 

97  163 

31 

30 

54  433 

57  274 

42  726 

97  159 

30 

31 

54  466 

57  312 

42  688 

97  154 

29 

32 

54  500 

57  351 

42  649 

97  149 

28 

33 

54  534 

57  389 

42  611 

97  145 

27 

34 

54  567 

57  428 

42  572 

97  140 

26 

35 

54  601 

57  466 

42  534 

97  135 

25 

36 

54  635 

57  504 

42  496 

97  130 

24 

37 

54  668 

57  543 

42  457 

97  126 

23 

38 

54  702 

57  581 

42  419 

97  121 

22 

39 

54  735 

57  619 

42  381 

97  116 

21 

40 

54  769 

57  658 

42  342 

97  111 

20 

41 

54  802 

57  696 

42  304 

97  107 

19 

42 

54  836 

57  734 

42  266 

97  102 

18 

43 

54  869 

57  772 

42  228 

97  097 

17 

44 

54  903 

57  810 

42  190 

97  092 

16 

45 

54  936 

57  849 

42  151 

97  087 

15 

46 

54  969 

57  887 

42  113 

97  083 

14 

47 

55  003 

57  925 

42  075 

97  078 

13 

48 

55  036 

57  963 

42  037 

97  073 

12 

49 

55  069 

58  001 

41  999 

97  068 

11 

50 

55  102 

58  039 

41961 

97  063 

10 

51 

55  136 

58  077 

41  923 

97  059 

9 

52 

55  169 

58  115 

41885 

97  054 

8 

53 

55  202 

58  153 

41847 

97  049 

7 

54 

55  235 

58  191 

41809 

97  044 

6 

55 

55  268 

58  229 

41  771 

97  039 

5 

56 

55  301 

58  267 

41  733 

97  035 

4 

57 

55  334 

58  304 

41  696 

97  030 

3 

58 

55  367 

58  342 

41  658 

97  025 

2 

59 

55  400 

58  380 

41620 

97  020 

1 

60 

55  433 

0 

58  418 

— 9 

log  cot 

41  582 
—10— 
log  tan 

97  015 

9— 

log  sin 

0 

t 

log  cos 

r 

70 


69 


38 

21° 

t 

log  sin 

n 

log  tan 

Q 

log  cot 

1  (\ 

log  cos 
n 

t 

0 

55  433 

58  418 

— iu — 

41  582 

97  015 

60 

1 

55  466 

58  455 

41545 

97  010 

59 

2 

55  499 

58  493 

41507 

97  005 

58 

3 

55  532 

58  531 

41469 

97  001 

57 

4 

55  564 

58  569 

41431 

96  996 

56 

5 

55  597 

58  606 

41394 

96  991 

55 

6 

55  630 

58  644 

41356 

96  986 

54 

7 

55  663 

58  681 

41319 

96  981 

53 

8 

55  695 

58  719 

41281 

96  976 

52 

9 

55  728 

58  757 

41243 

96  971 

51 

10 

55  761 

58  794 

41206 

96  966 

50 

11 

55  793 

58  832 

41  168 

96  962 

49 

12 

55  826 

58  869 

41  131 

96  957 

48 

13 

55  858 

58  907 

41093 

96  952 

47 

14 

55  891 

58  944 

41056 

96  947 

46 

15 

55  923 

58  981 

41019 

96  942 

45 

16 

55  956 

59  019 

40  981 

96  937 

44 

17 

55  988 

59  056 

40  944 

96  932 

43 

18 

56  021 

59  094 

40  906 

96  927 

42 

19 

56  053 

59  131 

40  869 

96  922 

41 

20 

56  085 

59  168 

40  832 

96  917 

40 

21 

56  118 

59  205 

40  795 

96  912 

39 

22 

56  150 

59  243 

40  757 

96  907 

38 

23 

56  182 

59  280 

40  720 

96  903 

37 

24 

56  21£ 

59  317 

40  683 

96  898 

36 

25 

56  247 

59  354 

40  646 

96  893 

35 

26 

56  279 

59  391 

40  609 

96  888 

34 

27 

56  311 

59  429 

40  571 

96  883 

33 

28 

56  343 

59  466 

40  534 

96  878 

32 

29 

56  375 

59  503 

40  497 

96  873 

31 

30 

56  408 

59  540 

40  460 

96  868 

30 

31 

56  440 

59  577 

40  423 

96  863 

29 

32 

56  472 

59  614 

40  386 

96  858 

28 

33 

56  504 

59  651 

40  349 

96  853 

27 

34 

56  536 

59  688 

40  312 

96  848 

26 

35 

56  568 

59  72£ 

40  275 

96  843 

25 

36 

56  599 

59  762 

40  238 

96  838 

24 

37 

56  631 

59  799 

40  201 

96  833 

23 

38 

56  663 

59  835 

40  165 

96  828 

22 

39 

56  695 

59  872 

40  128 

96  823 

21 

40 

56  727 

59  909 

40  091 

96  818 

20 

41 

56  759 

59  946 

40  054 

96  813 

19 

42 

56  790 

59  983 

40  017 

96  808 

18 

43 

56  822 

60  019 

39  981 

96  803 

17 

44 

56  854 

60  056 

39  944 

96  798 

16 

45 

56  886 

60  093 

39  907 

96  793 

15 

46 

56  917 

60  130 

39  870 

96  788 

14 

47 

56  949 

60  166 

39  834 

96  783 

13 

48 

56  980 

60  203 

39  797 

96  778 

12 

49 

57  012 

60  240 

39  760 

96  772 

11 

50 

57  044 

60  276 

39  724 

96  767 

10 

51 

57  075 

60  313 

39  687 

96  762 

9 

52 

57  107 

60  349 

39  651 

96  757 

8 

53 

57  138 

60  386 

39  614 

96  752 

7 

54 

57  169 

60  422 

39  578 

96  747 

6 

55 

57  201 

60  459 

39  541 

96  742 

5 

56 

57  232 

60  495 

39  505 

96  737 

4 

57 

57  264 

60  532 

39  468 

96  732 

3 

58 

57  295 

60  568 

39  432 

96  727 

2 

59 

57  326 

60  605 

39  395 

96  722 

1 

60 

f 

57  358 
log  cos 

60  641 
9 — 

log  cot 

39  359 
in 

96  717 

Q 

0 

— i\j — 

log  tan 

y 

log  sin 

r 

22 


r 

log  sin 

Q 

log  tan 

Q 

log  cot 
i  n 

log  cos 

t 

0 

57  358 

60  641 

— IU — 

39  359 

96  717 

60 

1 

57  389 

60  677 

39  323 

96  711 

59 

2 

57  420 

60  714 

39  286 

96  706 

58 

3 

57  451 

60  750 

39  250 

96  701 

57 

4 

57  482 

60  786 

39  214 

96  696 

56 

5 

57  514 

60  823 

39  177 

96  691 

55 

6 

57  545 

60  859 

39  141 

96  686 

54 

7 

SI  SI 6 

60  895 

39  105 

96  681 

53 

8 

57  607 

60  931 

39  069 

96  676 

52 

9 

57  638 

60  967 

39  033 

96  670 

51 

10 

57  669 

61004 

38  996 

96  665 

50 

11 

57  700 

61  040 

38  960 

96  660 

49 

12 

57  731 

61076 

38  924 

96  655 

48 

13 

57  762 

61  112 

38  888 

96  650 

47 

14 

57  793 

61  148 

38  852 

96  645 

46 

15 

57  824 

61  184 

38  816 

96  640 

45 

16 

57  855 

61220 

38  780 

96  634 

44 

17 

57  885 

61256 

38  744 

96  629 

43 

18 

57  916 

61  292 

38  708 

96  624 

42 

19 

57  947 

61328 

38  672 

96  619 

41 

20 

57  978 

61364 

38  636 

96  614 

40 

21 

58  008 

61400 

38  600 

96  608 

39 

22 

58  039 

61436 

38  564 

96  603 

38 

23 

58  070 

61472 

38  528 

96  598 

37 

24 

58  101 

61508 

38  492 

96  593 

36 

25 

58  131 

61544 

38  456 

96  588 

35 

26 

58  162 

61579 

38  421 

96  582 

34 

27 

58  192 

61615 

38  385 

96  577 

33 

28 

58  223 

61651 

38  349 

96  572 

32 

29 

58  253 

61687 

38  313 

96  567 

31 

30 

58  284 

61722 

38  278 

96  562 

30 

31 

58  314 

61  758 

38  242 

96  556 

29 

32 

58  345 

61  794 

38  206 

96  551 

28 

33 

58  375 

61830 

38  170 

96  546 

27 

34 

58  406 

61865 

38  135 

96  541 

26 

35 

58  436 

61901 

38  099 

96  535 

25 

36 

58  467 

61936 

38  064 

96  530 

24 

37 

58  497 

61972 

38  028 

96  525 

23 

38 

58  527 

62  008 

37  992 

96  520 

22 

39 

58  557 

62  043 

37  957 

96  514 

21 

40 

58  588 

62  079 

37  921 

96  509 

20 

41 

58  618 

62  114 

37  886 

96  504 

19 

42 

58  648 

62  150 

37  850 

96  498 

18 

43 

58  678 

62  185 

37  815 

96  493 

17 

44 

58  709 

62  221 

37  779 

96  488 

16 

45 

58  739 

62  256 

37  744 

96  483 

15 

46 

58  769 

62  292 

37  708 

96  477 

14 

47 

58  799 

62  327 

37  673 

96  472 

13 

48 

58  829 

62  362 

37  638 

96  467 

12 

49 

58  859 

62  398 

37  602 

96  461 

11 

50 

58  889 

62  433 

37  567 

96  456 

10 

51 

58  919 

62  468 

37  532 

96  451 

9 

52 

58  949 

62  504 

37  496 

96  445 

8 

53 

58  979 

62  539 

37  461 

96  440 

7 

54 

59  009 

62  574 

37  426 

96  435 

6 

55 

59  039 

62  609 

37  391 

96  429 

5 

56 

59  069 

62  645 

37  355 

96  424 

4 

57 

59  098 

62  680 

37  320 

96  419 

3 

58 

59  128 

62  715 

37  285 

96  413 

2 

59 

59  158 

62  750 

37-  250 

96  408 

1 

60 

59  188 
o 

62  785 

Q 

37  215 
Lin 

96  403 

9 

log  sin 

0 

f 

f 

y 

log  cos 

y 

log  cot 

log  tan 

68( 


67 


23 


r 

log  sin 

log  tan 

log  cot 

log  cos 

r 

Q 

Q 

— 10 — 

37  215 

Q 

0 

59  188 

62  785 

96  403 

60 

1 

59  218 

62  820 

37  180 

96  397 

59 

2 

59  247 

62  855 

37  145 

96  392 

58 

3 

59  277 

62  890 

37  110 

96  387 

57 

4 

59  307 

62  926 

37  074 

96  381 

56 

5 

59  336 

62  961 

37  039 

96  376 

55 

6 

59  366 

62  996 

37  004 

96  370 

54 

7 

59  396 

63  031 

36  969 

96  365 

53 

8 

59  425 

63  066 

36  934 

96  360 

52 

9 

59  453 

63  101 

36  899 

96  354 

51 

10 

59  484 

63  135 

36  865 

96  349 

50 

11 

59  514 

63  170 

36  830 

96  343 

49 

12 

59  543 

63  205 

36  795 

96  338 

48 

13 

59  573 

63  240 

36  760 

96  333 

47 

14 

59  602 

63  275 

36  725 

96  327 

46 

15 

59  632 

63  310 

36  690 

96  322 

45 

16 

59  661 

63  345 

36  655 

96  316 

it 

17 

59  690 

63  379 

36  621 

96  311 

& 

18 

59  720 

63  414 

36  586 

96  305 

42 

19 

59  749 

63  449 

36  551 

96  300 

41 

20 

59  778 

63  484 

36  516 

96  294 

40 

21 

59  808 

63  519 

36  481 

96  289 

39 

22 

59  837 

63  553 

36  447 

96  284 

38 

23 

59  866 

63  588 

36  412 

96  278 

37 

24 

59  895 

63  623 

36  377 

96  273 

36 

25 

59  924 

63  657 

36  343 

96  267 

35 

26 

59  954 

63  692 

36  308 

96  262 

34 

27 

59  983 

63  726 

36  274 

96  256 

33 

28 

60  012 

63  761 

36  239 

96  251 

32 

29 

60  041 

63  796 

36  204 

96  245 

31 

30 

60  070 

63  830 

36  170 

96  240 

30 

31 

60  099 

63  865 

36  135 

96  234 

29 

32 

60  128 

63  899 

36  101 

96  229 

28 

33 

60  157 

63  934 

36  066 

96  223 

27 

34 

60  186 

63  968 

36  032 

96  218 

26 

35 

60  215 

64  003 

35  997 

96  212 

25 

36 

60  244 

64  037 

35  963 

96  207 

24 

37 

60  273 

64  072 

35  928 

96  201 

23 

38 

60  302 

64  106 

35  894 

96  196 

22 

39 

60  331 

64  140 

35  860 

96  190 

21 

40 

60  359 

64  175 

35  825 

96  185 

20 

41 

60  388 

64  209 

35  791 

96  179 

19 

42 

60  417 

64  243 

35  757 

96  174 

18 

43 

60  446 

'64  278 

35  722 

96  168 

17 

44 

60  474 

64  312 

35  688 

96  162 

16 

45 

60  503 

64  346 

35  654 

96  157 

15 

46 

60  532 

64  381 

35  619 

96  151 

14 

47 

60  561 

64  415 

35  585 

96  146 

13 

48 

60  589 

64  449 

35  551 

96  140 

12 

49 

60  618 

64  483 

35  517 

96  135 

11 

50 

60  646 

64  517 

35  483 

96  129 

10 

51 

60  675 

64  552 

35  448 

96  123 

9 

52 

60  704 

64  586 

35  414 

96  118 

8 

53 

60  732 

64  620 

35  380 

96  112 

7 

54 

60  761 

64  654 

35  346 

96  107 

6 

55 

60  789 

64  688 

35  312 

96  101 

5 

56 

60  818 

64  722 

35  278 

96  095 

4 

57 

60  846 

64  756 

35  244 

96  090 

3 

58 

60  875 

64  790 

35  210 

96  084 

2 

59 

60  903 

64  824 

35  176 

96  079 

1 

60 

60  931 

Q 

64  858 

9 

log  cot 

35  142 
in 

96  073 

0 

r 

<3 

log  COS 

XV 

log  tan 

y 
log  sin 

f 

24° 

39 

r 

log  sin 
•  g 

log  tan 

9 

64  858 

log  cot 
— 10— 
35  142 

log  cos 

9 

96  073 

r 

0 

60  931 

60 

1 

60  960 

64  892 

35  108 

96  067 

59 

2 

60  988 

64  926 

35  074 

96  062 

58 

3 

61016 

64  960 

35  040 

96  056 

57 

4 

61045 

64  994 

35  006 

96  050 

56 

5 

61073 

65  028 

34  972 

96  045 

55 

6 

61  101 

65  062 

34  938 

96  039 

54 

7 

61  129 

65  096 

34  904 

96  034 

53 

8 

61  158 

65  130 

34  870 

96  028 

52 

9 

61  186 

65  164 

34  836 

96  022 

51 

10 

61214 

65  197 

34  803 

96  017 

50 

11 

61242 

65  231 

34  769 

96  011 

49 

12 

61  270 

65  265 

34  735 

96  005 

48 

13 

61298 

65  299 

34  701 

96  000 

47 

14 

61326 

65  333 

34  667 

95  994 

46 

15 

61354 

65  366 

34  634 

95  988 

45 

16 

61382 

65  400 

34  600 

95  982 

44 

17 

61411 

65  434 

34  566 

95  977 

43 

18 

61438 

65  467 

34  533 

95  971 

42 

19 

61466 

65  501 

34  499 

95  965 

41 

20 

61494 

65  535 

34  465 

95  960 

40 

21 

61  522 

65  568 

34  432 

95  954 

39 

22 

61  550 

65  602 

34  398 

95  948 

38 

23 

61578 

65  636 

34  364 

95  942 

37 

24 

61606 

65  669 

34  331 

95  937 

36 

25 

61634 

65  703 

34  297 

95  931 

35 

26 

61662 

65  736 

34  264 

95  925 

34 

27 

61689 

65  770 

34  230 

95  920 

33 

28 

61717 

65  803 

34  197 

95  914 

32 

29 

61  745 

65  837 

34  163 

95  908 

31 

30 

61773 

65  870 

34  130 

95  902 

30 

31 

61800 

65  904 

34  096 

95  897 

29 

32 

61828 

65  937 

34  063 

95  891 

28 

33 

61856 

65  971 

34  029 

95  885 

27 

34 

61883 

66  004 

33  996 

95  879 

26 

35 

61911 

66  038 

33  962 

95  873 

25 

36 

61939 

66  071 

33  929 

95  868 

24 

37 

61966 

66  104 

33  896 

95  862 

23 

38 

61  994 

66  138 

33  862 

95  856 

22 

39 

62  021 

66  171 

33  829 

95  850 

21 

40 

62  049 

66  204 

33  796 

95  844 

20 

41 

62  076 

66  238 

33  762 

95  839 

19 

42 

62  104 

66  271 

33  729 

95  833 

18 

43 

62  131 

66  304 

33  696 

95  827 

17 

44 

62  159 

66  337 

33  663 

95  821 

16 

45 

62  186 

66  371 

33  629 

95  815 

15 

46 

62  214 

66  404 

33  596 

95  810 

14 

47 

62  241 

66  437 

33  563 

95  804 

13 

48 

62  268 

66  470 

33  530 

95  798 

12 

49 

62  296 

66  503 

33  497 

95  792 

11 

50 

62  323 

66  537 

33  463 

95  786 

10 

51 

62  350 

66  570 

33  430 

95  780 

9 

52 

62  377 

66  603 

33  397 

95  775 

8 

53 

62  405 

66  636 

33  364 

95  769 

7 

54 

62  432 

66  669 

33  331 

95  763 

6 

55 

62  459 

66  702 

33  298 

95  757 

5 

56 

62  486 

66  735 

33  265 

95  751 

4 

57 

62  513 

66  768 

33  232 

95  745 

3 

58 

62  541 

66  801 

33  199 

95  739 

2 

59 

62  568 

66  834 

33  166 

95  733 

1 

60 

62  595 

66  867 

— 9 — 

log  cot 

33  133 

95  728 

0 

f 

log  COS 

— iu » — 

log  tan  leg  sin 

' 

66l 


65( 


40 

25° 

r 

log  sin 

n 

log  tan 

9 

66  867 

log  cot 
—10— 
33  133 

log  cos 

f 

0 

62  595 

95  728 

60 

1 

62  622 

66  900 

33  100 

95  722 

59 

2 

62  649 

66  933 

33  067 

95  716 

58 

3 

62  676 

66  966 

33  034 

95  710 

57 

4 

62  703 

66  999 

33  001 

95  704 

56 

5 

62  730 

67  032 

32  968 

95  698 

55 

6 

62  757 

67  065 

32  935 

95  692 

54 

7 

62  784 

67  098 

32  902 

95  686 

53 

8 

62  811 

67  131 

32  869 

95  680 

52 

9 

62  838 

67  163 

32  837 

95  674 

51 

10 

62  865 

67  196 

32  804 

95  668 

50 

11 

62  892 

67  229 

32  771 

95  663 

49 

12 

62  918 

67  262 

32  738 

95  657 

48 

13 

62  945 

67  295 

32  705 

95  651 

47 

14 

62  972 

67  327 

32  673 

95  645 

46 

15 

62  999 

67  360 

32  640 

95  639 

45 

16 

63  026 

67  393 

32  607 

95  633 

44 

17 

63  052 

67  426 

32  574 

95  627 

43 

18 

63  079 

67  458 

32  542 

95  621 

42 

19 

63  106 

67  491 

32  509 

95  615 

41 

20 

63  133 

67  524 

32  476 

95  609 

40 

21 

63  159 

67  556 

32  444 

95  603 

39 

22 

63  186 

67  589 

32  411 

95  597 

38 

23 

63  213 

67  622 

32  378 

95  591 

37 

24 

63  239 

67  654 

32  346 

95  585 

36 

25 

63  266 

67  687 

32  313 

95  579 

35 

26 

63  292 

67  719 

32  281 

95  573 

34 

27 

63  319 

67  752 

32  248 

95  567 

33 

28 

63  345 

67  785 

32  21 5 

95  561 

32 

29 

63  372 

67  817 

32  183 

95  555 

31 

30 

63  398 

67  850 

32  150 

95  549 

30 

31 

63  425 

67  882 

32  118 

95  543 

29 

32 

63  451 

67  915 

32  085 

95  537 

28 

33 

63  478 

67  947 

32  053 

95  531 

27 

34 

63  504 

67  980 

32  020 

95  525 

26 

35 

63  531 

68  012 

31988 

95  519 

25 

36 

63  557 

68  044 

31956 

95  513 

24 

37 

63  583 

68  077 

31923 

95  507 

23 

38 

63  610 

68  109 

31891 

95  500 

22 

39 

63  636 

68  142 

31858 

95  494 

21 

40 

63  662 

68  174 

31826 

95  488 

20 

41 

63  689 

68  206 

31  794 

95  482 

19 

42 

63  715 

68  239 

31  761 

95  476 

18 

43 

63  741 

68  271 

31729 

95  470 

17 

44 

63  767 

68  303 

31697 

95  464 

16 

45 

63  794 

68  336 

31664 

95  458 

15 

46 

63  820 

68  368 

31632 

95  452 

14 

47 

63  846 

68  400 

31  600 

95  446 

13 

48 

63  872 

68  432 

31568 

95  440 

12 

49 

63  898 

68  465 

31535 

95  434 

11 

50 

63  924 

68  497 

31503 

95  427 

10 

51 

63  950 

68  529 

31471 

95  421 

9 

52 

63  976 

68  561 

31439 

95  415 

8 

53 

64  002 

68  593 

31407 

95  409 

7 

54 

64  028 

68  626 

31  374 

95  403 

6 

55 

64  054 

68  658 

31342 

95  397 

5 

56 

64  080 

68  690 

31310 

95  391 

4 

57 

64  106 

68  722 

31  278 

95  384 

3 

58 

64  132 

68  754 

31246 

95  378 

2 

59 

64  158 

68  786 

31214 

95  372 

1 

60 

64  184 

68  818 

31182 

95  366 

0 

Q 

Q 

10 

o 

r 

log  COS 

log  cot 

log  tan 

log  sin 

t 

26° 


f 

log  sin 

9 

64  184 

log  tan 

9 

68  818 

log  cot 
—10— 
31  182 

log  cos 

f 

0 

95  366 

60 

1 

64  210 

68  850 

31  150 

95  360 

59 

2 

64  236 

68  882 

31  118 

95  354 

58 

3 

64  262 

68  914 

31086 

95  348 

57 

4 

64  288 

68  946 

31054 

95  341 

56 

5 

64  313 

68  978 

31022 

95  335 

55 

6 

64  339 

69  010 

30  990 

95  329 

54 

7 

64  365 

69  042 

30  958 

95  323 

53 

8 

64  391 

69  074 

30  926 

95  317 

52 

9 

64  417 

69  106 

30  894 

95  310 

51 

10 

64  442 

69  138 

30  862 

95  304 

50 

11 

64  468 

69  170 

30  830 

95  298 

49 

12 

64  494 

69  202 

30  798 

95  292 

48 

13 

64  519 

69  234 

30  766 

95  286 

47 

14 

64  545 

69  266 

30  734 

95  279 

46 

15 

64  571 

69  298 

30  702 

95  273 

45 

V 

64  596 

69  329 

30  671 

95  267 

44 

17 

64  622 

69  361 

30  639 

95  261 

43 

18 

64  647 

69  393 

30  607 

95  254 

42 

19 

64  673 

69  425 

30  575 

95  248 

41 

20 

64  698 

69  457 

30  543 

95  242 

40 

21 

64  724 

69  488 

30  512 

95  236 

39 

22 

64  749 

69  520 

30  480 

95  229 

38 

23 

64  775 

69  552 

30  448 

95  223 

37 

24 

64  800 

69  584 

30  416 

95  217 

36 

25 

64  826 

69  615 

30  385 

95  211 

35 

26 

64  851 

69  647 

30  353 

95  204 

34 

27 

64  877 

69  679 

30  321 

95  198 

33 

28 

64  902 

69  710 

30  290 

95  192 

32 

29 

64  927 

69  742 

30  258 

95  185 

31 

30 

64  953 

69  774 

30  226 

95  179 

30 

31 

64  978 

69  805 

30  195 

95  173 

29 

32 

65  003 

69  837 

30  163 

95  167 

28 

33 

65  029 

69  868 

30  132 

95  160 

27 

34 

65  054 

69  900 

30  100 

95  154 

26 

35 

65  079 

69  932 

30  068 

95  148 

25 

36 

65  104 

69  963 

30  037 

95  141 

24 

37 

65  130 

69  995 

30  005 

95  135 

23 

38 

65  155 

70  026 

29  974 

95  129 

22 

39 

65  180 

70  058 

29  942 

95  122 

21 

40 

65  205 

70  089 

29  911 

95  116 

20 

41 

65  230 

70  121 

29  879 

95  110 

19 

42 

65  255 

70  152 

29  848 

95  103 

18 

43 

65  281 

70  184 

29  816 

95  097 

17 

44 

65  306 

70  215 

29  785 

95  090 

16 

45 

65  331 

70  247 

29  753 

95  084 

15 

46 

65  356 

70  278 

29  722 

95  078 

14 

47 

65  381 

70  309 

29  691 

95  071 

13 

48 

65  406 

70  341 

29  659 

95  065 

12 

49 

65  431 

70  372 

29  628 

95  059 

11 

50 

65  456 

70  404 

29  596 

95  052 

10 

51 

65  481 

70  435 

29  565 

95  046 

9 

52 

65  506 

70  466 

29  534 

95  039 

8 

53 

65  531 

70  498 

29  502 

95  033 

7 

54 

65  556 

70  529 

29  471 

95  027 

C 

55 

65  580 

70  560 

29  440 

95  020 

5 

56 

65  605 

70  592 

29  408 

95  014 

4 

57 

65  630 

70  623 

29  377 

95  007 

3 

58 

65  655 

70  654 

29  346 

95  001 

2 

59 

65  680 

70  685 

29  315 

94  995 

1 

60 

65  705 

70  717 

Q 

29  283 
in 

94  988 

Q 

0 

t 

a 

log  cos 

log  cot 

i-V 

log  tan 

y 

log  sin 

t 

64( 


63 


27( 


f 

log  sin 

log  tan 

log  cot 

log  cos 

r 

9 

9 

70  717 

— 10 — 

29  283 

g 

0 

65  705 

94  988 

60 

1 

65  729 

70  748 

29  252 

94  982 

59 

2 

65  754 

70  779 

29  221 

94  975 

58 

3 

65  779 

70  810 

29  190 

94  969 

57 

4 

65  804 

70  841 

29  159 

94  962 

56 

5 

65  828 

70  873 

29  127 

94  956 

55 

6 

65  853 

70  904 

29  096 

94  949 

54 

7 

65  878 

70  935 

29  065 

94  943 

53 

8 

65  902 

70  966 

29  034 

94  936 

52 

9 

65  927 

70  997 

29  003 

94  930 

51 

10 

65  952 

71028 

28  972 

94  923 

50 

11 

65  976 

71059 

28  941 

94  917 

49 

12 

66  001 

71090 

28  910 

94  911 

48 

13 

66  025 

71  121 

28  879 

94  904 

47 

14 

66  050 

71  153 

28  847 

94  898 

46 

15 

66  075 

71184 

28  816 

94  891 

45 

16 

66  099 

71215 

28  785 

94  885 

44 

17 

66  124 

71246 

28  754 

94  878 

43 

18 

66  148 

71277 

28  723 

94  871 

42 

19 

66173 

71308 

28  692 

94  865 

41 

20 

66  197 

71339 

28  661 

94  858 

40 

21 

66  221 

71370 

'28  630 

94  852 

39 

22 

66  246 

71401 

28  599 

94  845 

38 

23 

66  270 

71431 

28  569 

94  839 

37 

24 

66  295 

71462 

28  538 

94  832 

36 

25 

66  319 

71493 

28  507 

94  826 

35 

26 

66  343 

71524 

28  476 

94  819 

34 

27 

66  368 

71  555 

28  445 

94  813 

33 

28 

66  392 

71586 

28  414 

94  806 

32 

29 

66  416 

71617 

28  383 

94  799 

31 

30 

66  441 

71648 

28  352 

94  793 

30 

31 

66  465 

71679 

28  321 

94  786 

29 

32 

66  489 

71709 

28  291 

94  780 

28 

33 

66  513 

71740 

28  260 

94  773 

27 

34 

66  537 

71  771 

28  229 

94  767 

26 

35 

66  562 

71802 

28  198 

94  760 

25 

36 

66  586 

71833 

28  167 

94  753 

24 

37 

66  610 

71863 

28  137 

94  747 

23 

38 

66  634 

71894 

28  106 

94  740 

22 

39 

66  658 

7192i 

28  075 

94  734 

21 

40 

66  682 

71955 

28  045 

94  727 

20 

41 

66  706 

71986 

28  014 

94  720 

19 

42 

66  731 

72  017 

27  983 

94  714 

18 

43 

66  755 

72  048 

27  952 

94  707 

17 

44 

66  779 

72  078 

27  922 

94  700 

16 

45 

66  803 

72  109 

27  891 

94  694 

15 

46 

66  827 

72  140 

27  860 

94  687 

14 

47 

66  851 

72  170 

27  830 

94  680 

13 

48 

66  875 

72  201 

27  799 

94  674 

12 

49 

66  899 

72  231 

27  769 

94  667 

11 

50 

66  922 

72  262 

27  738 

94  660 

10 

51 

66  946 

72  293 

27  707 

94  654 

9 

52 

66  970 

72  323 

27  677 

94  647 

8 

53 

66  994 

72  354 

27  646 

94  640 

7 

54 

67  018 

72  384 

27  616 

94  634 

6 

55 

67  042 

72  415 

27  585 

94  627 

5 

56 

67  066 

72  445 

27  555 

94  620 

4 

57 

67  090 

72  476 

27  524 

94  614 

3 

58 

67  113 

72  506 

27  494 

94  607 

2 

59 

67  137 

72  537 

27  463 

94  600 

1 

60 

67  161 

72  567 

27  433 

94  593 

0 

—  9 — 

log  cos 

— 9— 
log  cot 

—10  — 

log  tan 

0 

t 

y 

log  sin 

t 

28° 

41 

r 

log  sin 
9 

log  tan 
9 

log  cot 
10 

log  cos 
9 

r 

0 

67  161 

72  567 

27  433 

94  593 

60 

1 

67  185 

72  598 

27  402 

94  587 

59 

2 

67  208 

72  628 

27  372 

94  580 

58 

3 

67  232 

72  659 

27  341 

94  573 

57 

4 

67  256 

72  689 

27  311 

94  567 

56 

5 

67  280 

.72  720 

27  280 

94  560 

55 

6 

67  303 

72  750 

27  250 

94  553 

54 

7 

67  327 

72  780 

27  220 

94  546 

53 

8 

67  350 

72  811 

27  189 

94  540 

52 

9 

67  374 

72  841 

27  159 

94  533 

51 

10 

67  398 

72  872 

27  128 

94  526 

50 

11 

67  421 

72  902 

27  098 

94  519 

49 

12 

67  445 

72  932 

27  068 

94  513 

48 

13 

67  468 

72  963 

27  037 

94  506 

47 

14 

67  492 

72  993 

27  007 

94  499 

46 

15 

67  515 

73  023 

26  977 

94  492 

45 

16 

67  539 

73  054 

26  946 

94  485 

44 

17 

67  562 

73  084 

26  916 

94  479 

43 

18 

67  586 

73  114 

26  886 

94  472 

42 

19 

67  609 

73  144 

26  856 

94  465 

41 

20 

67  633 

73  175 

26  825 

94  45S 

40 

21 

67  656 

73  205 

26  795 

94  451 

39 

22 

67  680 

73  235 

26  765 

94  445 

38 

23 

67  703 

73  265 

26  735 

94  438 

37 

24 

67  726 

73  295 

26  705 

94  431 

36 

25 

67  750 

73  326 

26  674 

94  424 

35 

26 

67  773 

73  356 

26  644 

94  417 

34 

27 

67  796 

73  3S6 

26  614 

94  410 

33 

28 

67  820 

73  416 

26  584 

94  404 

32 

29 

67  843 

73  446 

26  554 

94  397 

31 

30 

67  866 

73  476 

26  524 

94  390 

30 

31 

67  890 

73  507 

26  493 

94  383 

29 

32 

67  913 

73  537 

26  463 

94  376 

28 

33 

67  936 

73  567 

26  433 

94  369 

27 

34 

67  959 

73  597 

26  403 

94  362 

26 

35 

67  982 

73  627 

26  373 

94  355 

25 

36 

68  006 

73  657 

26  343 

94  349 

24 

37 

68  029 

73  687 

•26  313 

94  342 

23 

38 

68  052 

73  717 

26  283 

94  335 

22 

39 

68  075 

73  747 

26  253 

94  328 

21 

40 

68  098 

73  777 

26  223 

94  321 

20 

41 

68  121 

73  807 

26  193 

94  314 

19 

42 

68  144 

73  837 

26  163 

94  307 

18 

43 

68  167 

73  867 

26  133 

94  300 

17 

44 

68  190 

73  897 

26  103 

94  293 

16 

45 

68  213 

73  927 

26  073 

94  286 

15 

46 

68  237 

73  957 

26  043 

94  279 

14 

47 

68  260 

73  987 

26  013 

94  273 

13 

48 

68  283 

74  017 

25  983 

94  266 

12 

49 

68  305 

74  047 

25  953 

94  259 

11 

50 

68  328 

74  077 

25  923 

94  252 

10 

51 

68  351 

74  107 

25  893 

94  245 

9 

52 

68  374 

74  137 

25  863 

94  238 

8 

53 

68  397 

74  166 

25  834 

94  231 

7 

54 

68  420 

74  196 

25  804 

94  224 

6 

55 

68  443 

74  226 

25  774 

94  217 

5 

56 

68  466 

74  256 

25  744 

94  210 

4 

57 

68  489 

74  286 

25  714 

94  203 

3 

58 

68  512 

74  316 

25  684 

94  196 

2 

59 

68  534 

74  345 

25  65i 

94  189 

1 

60 

68  557 

74  375 

25  625 

94  182 

0 

o 

q 

in 

log  sin 

t 

log  cos 

V 

log  cot 

log  tan 

' 

62 


61 


42 

29° 

/ 

log  sin 

n 

log  tan 
o 

log  cot 
in 

log  cos 

Q , 

t 

0 

68  557 

y 
74  375 

— w — 

25  625 

y 

94  182 

60 

i 

68  580 

74  405 

25  595 

94  175 

59 

2 

68  603 

74  435 

25  565 

94  168 

58 

3 

68  625 

74  465 

25  535 

94  161 

57 

4 

68  648 

74  494 

25  506 

94  154 

56 

5 

68  671 

74  524 

25  476 

94  147 

55 

6 

68  694 

74  554 

25  446 

94  140 

54 

7 

68  716 

74  583 

25  417 

94  133 

53 

8 

68  739 

74  613 

25  387 

94  126 

52 

9 

68  762 

74  643 

25  357 

94  119 

51 

10 

68  784 

74  673 

25  327 

94  112 

50 

11 

68  807 

74  702 

25  298 

94  105 

49 

12 

68  829 

74  732 

25  268 

94  098 

48 

13 

68  852 

74  762 

25  238 

94  090 

47 

14 

68  875 

74  791 

25  209 

94  083 

46 

15 

68  897 

74  821 

25  179 

94  076 

45 

16 

68  920 

74  851 

25  149 

94  069 

44 

17 

68  942 

74  880 

25  120 

94  062 

43 

18 

68  965 

74  910 

25  090 

94  055 

42 

19 

68  987 

74  939 

25  061 

94  048 

41 

20 

69  010 

74  969 

25  031 

94  041 

40 

21 

69  032 

74  998 

25  002 

94  034 

39 

22 

69  055 

75  028 

24  972 

94  027 

38 

23 

69  077 

75  058 

24  942 

94  020 

37 

24 

69  100 

75  087 

24  913 

94  012 

36 

25 

69  122 

75  117 

24  883 

94  005 

35 

26 

69  144 

75  146 

24  854 

93  998 

34 

27 

69  167 

75  176 

24  824 

93  991 

33 

28 

69  189 

75  205 

24  795 

93  984 

32 

29 

69  212 

75  235 

24  765 

93  977 

31 

30 

69  234 

75  264 

24  736 

93  970 

30 

31 

69  256 

75  294 

24  706 

93  963 

29 

32 

69  279 

75  323 

24  677 

93  955 

28 

33 

69  301 

75  353 

24  647 

93  948 

27 

34 

69  323 

75  382 

24  618 

93  941 

26 

35 

69  345 

75  411 

24  589 

93  934 

25 

36 

69  368 

75  441 

24  559 

93  927 

24 

37 

69  390 

75  470 

24  530 

93  920 

23 

38 

69  412 

75  500 

24  500 

93  912 

22 

39 

69  434 

75  529 

24  471 

93  905 

21 

40 

69  456 

75  558 

24  442 

93  898 

20 

41 

69  479 

75  588 

24  412 

93  891 

19 

42 

69  501 

75  617 

24  383 

93  884 

18 

43 

69  523 

75  647 

24  353 

93  876 

17 

44 

69  545 

75  676 

24  324 

93  869 

16 

45 

69  567 

75  705 

24  295 

93  862 

15 

46 

69  589 

75  735 

24  265 

93  855 

14 

47 

69  611 

75  764 

24  236 

93  847 

13 

48 

69  633 

75  793 

24  207 

93  840 

12 

49 

69  655 

75  822 

24  178 

93  833 

11 

50 

69  677 

75  852 

24  148 

93  826 

10 

51 

69  699 

75  881 

24  119 

93  819 

9 

52 

69  721 

75  910 

24  090 

93  811 

8 

53 

69  743 

75  939 

24  061 

93  804 

7 

54 

69  765 

75  969 

24  031 

93  797 

6 

55 

69  787 

75  998 

24  002 

93  789 

5 

56 

69  809 

76  027 

23  973 

93  782 

4 

57 

69  831 

76  056 

23  944 

93  775 

3 

58 

69  853 

76  086 

23  914 

93  768 

2 

59 

69  875 

76  115 

23  885 

93  760 

1 

60 

69  897 

n 

76  144 

Q 

23  856 
in 

93  753 

0 

0 

t 

a 

log  cos 

y 

log  cot 

— iu  — 

log  tan 

y 

log  sin 

r 

30 

t 
0 

log  sin 

9 

69  897 

log  tan 

9 — 

76  144 

log  cot 
in 

log  cos 

Q 

r 

iu — 

23  856 

93  753 

60 

1 

69  919 

76' 173 

23  827 

93  746 

59 

2 

69  941 

76  202 

23  798 

93  738 

58 

3 

69  963 

76  231 

23  769 

93  731 

57 

4 

69  984 

76  261 

23  739 

93  724 

56 

5 

70  006 

76  290 

23  710 

93  717 

55 

6 

70  028 

76  319 

23  681 

93  709 

54 

7 

70  050 

76  348 

28  652 

93  702 

53 

8 

70  072 

76  377 

23  623 

93  695 

52 

9 

70  093 

76  406 

23  594 

93  687 

51 

10 

70  115 

76  435 

23  565 

93  680 

50 

11 

70  137 

76  464 

23  536 

93  673 

49 

12 

70  159 

76  493 

23  507 

93  665 

48 

13 

70  180 

76  522 

23  478 

93  658 

47 

14 

70  202 

76  551 

23  449 

93  650 

46 

15 

70  224 

76  580 

23  420 

93  643 

45 

16 

70  245 

76  609 

23  391 

93  636 

44 

17 

70  267 

76  639 

23  361 

93  628 

43 

18 

70  288 

76  668 

23  332 

93  621 

42 

19 

70  310 

76  697 

23  303 

93  614 

41 

20 

70  332 

76  725 

23  275 

93  606 

40 

21 

70  353 

76  754 

23  246 

93  599 

39 

22 

70  375 

76  783 

23  217 

93  591 

38 

23 

70  396 

76  812 

23  188 

93  584 

37 

24 

70  418 

76  841 

23  159 

93  577 

36 

25 

70  439 

76  $70 

23  130 

93  569 

35 

26 

70  461 

76  899 

23  101 

93  562 

34 

27 

70  482 

76  928 

23  072 

93  554 

33 

28 

70  504 

76  957 

23  043 

93  547 

32 

29 

70  525 

76  986 

23  014 

93  539 

31 

30 

70  547 

77  015 

22  985 

93  532 

30 

31 

70  568 

77  044 

22  956 

93  525 

29 

32 

70  590 

77  073 

22  927 

93  517 

28 

33 

70  611 

77  101 

22  899 

93  510 

27 

34 

70  633 

77  130 

22  870 

93  502 

26 

35 

70  654 

77  159 

22  841 

93  495 

25 

36 

70  675 

77  188 

22  812 

93  487 

24 

37 

70  697 

77  217 

22  783 

93  480 

23 

38 

70  718 

77  246 

22  754 

93  472 

22 

39 

70  739 

77  274 

22  726 

93  465 

21 

40 

70  761 

77  303 

22  697 

93  457 

20 

41 

70  782 

77  332 

22  668 

93  450 

19 

42 

70  803 

77  361 

22  639 

93  442 

18 

43 

70  824 

77  390 

22  610 

93  435 

17 

44 

70  846 

77  418 

22  582 

93  427 

16 

45 

70  867 

77  447 

22  553 

93  420 

15 

46 

70  888 

77  476 

22  524 

93  412 

14 

47 

70  909 

77  505 

22  495 

93  405 

13 

48 

70  931 

77  533 

22  467 

93  397 

12 

49 

70  952 

77  562 

22  438 

93  390 

11 

50 

70  973 

77  591 

22  409 

93  382 

10 

51 

70  994 

77  619 

22  381 

93  375 

9 

52 

71015 

77  648 

22  352 

93  367 

8 

53 

71  036 

77  677 

22  323 

93  360 

7 

54 

71058 

77  706 

22  294 

93  352 

6 

55 

71079 

77  734 

22  266 

93  344 

5 

56 

71  100 

77  763 

22  237 

93  337 

4 

57 

71  121 

77  791 

22  209 

93  329 

3 

58 

71  142 

77  820 

22  180 

93  322 

2 

59 

71  163 

77  849 

22  151 

93  314 

1 

60 

71  184 

0 

77  877 

n 

22  123 
in 

93  307 
o 

0 

f 

y 

log  cos 

y 

log  cot 

— iu 

log  tan 

y 

log  sin 

t 

60 


59 


31 


f 

log  sin 

log  tan 

log  cot 

log  cos 

r 

Q 

9 

77  877 

—  10 — 
22  123 

9  ■■ 

0 

71  184 

93  307 

60 

1 

71  205 

77  906 

22  094 

93  299 

59 

2 

71  226 

77  935 

22  065 

93  291 

58 

3 

71247 

77  963 

22  037 

93  284 

57 

4 

71268 

77  992 

22  008 

93  276 

56 

5 

71289 

78  020 

21980 

93  269 

55 

6 

71310 

78  049 

21951 

93  261 

54 

7 

71331 

78  077 

21923 

93  253 

53 

8 

71352 

78  106 

21  894 

93  246 

52 

9 

71373 

78  135 

21  865 

93  238 

51 

10 

71393 

78  163 

21  837 

93  230 

50 

11 

71414 

78  192 

21808 

93  223 

49 

12 

71435 

78  220 

21  780 

93  215 

48 

13 

71456 

78  249 

21  751 

93  207 

47 

14 

71477 

78  277 

21  723 

93  200 

46 

15 

71498 

78  306 

21694 

93  192 

45 

16 

71519 

78  334 

21666 

93  184 

44 

17 

71  539 

78  363 

21637 

93  177 

43 

18 

71  560 

78  391 

21609 

93  169 

42 

19 

71  581 

78  419 

21  581 

93  161 

41 

20 

71602 

78  448 

21552 

93  154 

40 

21 

71  622 

78  476 

21  524 

93  146 

39 

22 

71643 

78  505 

21  495 

93  138 

38 

23 

71664 

78  533 

21467 

93  131 

37 

24 

71685 

78  562 

21438 

93  123 

36 

25 

71  705 

78  590 

2L410 

93  115 

35 

26 

71  726 

78  618 

21382 

93  108 

34 

27 

71747 

78  647 

21353 

93  100 

33 

28 

71  767 

78  675 

21325 

93  092 

32 

29 

71788 

78  704 

21296 

93  084 

31 

30 

71809 

78  732 

21268 

93  077 

30 

31 

71829 

78  760 

21240 

93  069 

29 

32 

71850 

78  789 

21  211 

93  061 

28 

33 

71870 

78  817 

21  183 

93  053 

27 

34 

71891 

78  845 

21155 

93  046 

26 

35 

71911 

78  874 

21  126 

93  038 

25 

36 

71932 

78  902 

21098 

93  030 

24 

37 

71952 

78  930 

21070 

93  022 

23 

38 

71973 

78  959 

21  041 

93  014 

22 

39 

71994 

78  987 

21013 

93  007 

21 

40 

72  014 

79  015 

20  985 

92  999 

20 

41 

72  034 

79  043 

20  957 

92  991 

19 

42 

72  055 

79  072 

20  928 

92  983 

18 

43 

72  075 

79  100 

20  900 

92  976 

17 

44 

72  096 

79  128 

20  872 

92  968 

16 

45 

72  116 

79  156 

20  844 

92  960 

15 

46 

72  137 

79  185 

20  815 

92  952 

14 

47 

72  157 

79  213 

20  787 

92  944 

13 

48 

72  177 

79  241 

20  759 

92  936 

12 

49 

72  198 

79  269 

20  731 

92  929 

11 

50 

72  218 

79  297 

20  703 

92  921 

10 

51 

72  238 

79  326 

20  674 

92  913 

9 

52 

72  259 

79  354 

20  646 

92  905 

8 

53 

72  279 

79  382 

20  618 

92  897 

7 

54 

72  299 

79  410 

20  590 

92  889 

6 

55 

72  320 

79  438 

20  562 

92  881 

5 

56 

72  340 

79  466 

20  534 

92  874 

4 

57 

72  360 

79  495 

20  505 

92  866 

3 

58 

72  381 

79  523 

20  477 

92  858 

2 

59 

72  401 

79  551 

20  449 

92  850 

1 

60 

72  421 

79  579 

9 

log  cot 

20  421 
—10  — 

log  tan 

92  842 
9 

0 

f 

log  cos 

log  sin 

r 

3 

2° 

43 

t 

log  sin 

log  tan 

log  cot 

log  cos 

t 

g 

10 

9 

0 

72  421 

79  579 

20  421 

92  842 

60 

1 

72  441 

79  607 

20  393 

92  834 

59 

2 

72  461 

79  635 

20  365 

92  826 

58 

3 

72  482 

79  663 

20  337 

92  818 

57 

4 

72  502 

79  691 

20  309 

92  810 

56 

5 

72  522 

79  719 

20  281 

92  803 

55 

6 

72  542 

79  747 

20  253 

92  795 

54 

7 

72  562 

79  776 

20  224 

92  787 

53 

8 

72  582 

79  804 

20  196 

92  779 

52 

9 

72  602 

79  832 

20  168 

92  771 

51 

10 

72  622 

79  860 

20  140 

92  763 

50 

11 

72  643 

79  888 

20  112 

92  755 

49 

12 

72  663 

79  916 

20  084 

92  747 

48 

13 

72  683 

79  944 

20  056 

92  739 

47 

14 

72  703 

79  972 

20  028 

92  731 

46 

15 

72  723 

80  000 

20  000 

92  723 

45 

16 

72  743 

80  028 

19  972 

92  715 

44 

17 

72  763 

80  056 

19  944 

92  707 

43 

18 

72  783 

80  084 

19  916 

92  699 

42 

19 

72  803 

80  112 

19  888 

92  691 

41 

20 

72  823 

80  140 

19  860 

92  683 

40 

21 

72  843 

80  168 

19  832 

92  675 

39 

22 

72  863 

80  195 

19  805 

92  667 

38 

23 

72  883 

80  223 

19  777 

92  659 

37 

24 

72  902 

80  251 

19  749 

92  651 

36 

25 

72  922 

80  279 

19  721 

92  643 

35 

26 

72  942 

80  307 

19  693 

92  635 

34 

27 

72  962 

80  335 

19  665 

92  627 

33 

28 

72  982 

80  363 

19  637 

92  619 

32 

29 

73  002 

80  391 

19  609 

92  611 

31 

30 

73  022 

80  419 

19  581 

92  603 

30 

31 

73  041 

80  447 

19  553 

92  595 

29 

32 

73  061 

80  474 

19  526 

92  587 

28 

33 

73  081 

80  502 

19  498 

92  579 

27 

34 

73  101 

80  530 

19  470 

92  571 

26 

35 

73  121 

80  558 

19  442 

92  563 

25 

36 

73  140 

80  586 

19  414 

92  555 

24 

37 

73  160 

80  614 

19  386 

92  546 

23 

38 

73  180 

80  642 

19  358 

92  538 

22 

39 

73  200 

80  669 

19  331 

92  530 

21 

40 

73  219 

80  697 

19  303 

92  522 

20 

41 

73  239 

80  725 

19  275 

92  514 

19 

42 

73  259 

80  753 

19  247 

92  506 

18 

43 

73  278 

80  781 

19  219 

92  498 

17 

44 

73  298 

80  808 

19  192 

92  490 

16 

45 

73  318 

80  836 

19  164 

92  482 

15 

46 

73  337 

80  864 

19  136 

92  473 

14 

47 

73  357 

80  892 

19  108 

92  465 

13 

48 

73  377 

80  919 

19  081 

92  457 

12 

49 

73  396 

80  947 

19  053 

92  449 

11 

50 

73  416 

80  975 

19  025 

92  441 

10 

51 

73  435 

81003 

18  997 

92  433 

9 

52 

73  455 

81030 

18  970 

92  425 

8 

53 

73  474 

81058 

18  942 

92  416 

7 

54 

73  494 

81086 

18  914 

92  408 

6 

55 

73  513 

81  113 

18  887 

92  400 

5 

56 

73  533 

81  141 

18  859 

92  392 

4 

57 

73  552 

81  169 

18  831 

92  384 

3 

58 

73  572 

81  196 

18  804 

92  376 

2 

59 

73  591 

81224 

18  776 

92  367 

1 

60 

73  611 
— 9 — 

log  cos 

81252 

9 

log  cot 

18  748 
—10— 
log  tan 

92  359 
9 

0 

r 

log  sin 

r 

58 


57 


44 

33° 

t 

log  sin  log  tan 

q 1    n 

log  cot 
m 

log  cos 

Q 

; 

0 

73  611 

81  252 

— iu — 

18  748 

92  359 

60 

1 

73  630 

81279 

18  721 

92  351 

59 

2 

73  650 

81307 

18  693 

92  343 

58 

3 

73  669 

81335 

18  665 

92  335 

57 

4 

73  689 

81  362 

18  638 

92  326 

56 

5 

73  708 

81390 

18  610 

92  318 

55 

6 

73  727 

81418 

18  582 

92  310 

54 

7 

73  747 

81445 

18  555 

92  302 

53 

8 

73  766 

81473 

18  527 

92  293 

52 

9 

73  785 

81  500 

18  500 

92  285 

51 

10 

73  805 

81528 

18  472 

92  277 

50 

11 

73  824 

81  556 

18  444 

92  269 

49 

12 

73  843 

81  583 

18  417 

92  260 

48 

13 

73  863 

81611 

18  389 

92  252 

47 

14 

73  882 

81638 

18  362 

92  244 

46 

15 

73  901 

81  666 

18  334 

92  235 

45 

16 

73  921 

81693 

18  307 

92  227 

44 

17 

73  940 

81  721 

18  279 

92  219 

43 

18 

73  959 

81  748 

18  252 

92  211 

42 

19 

73  978 

81  776 

18  224 

92  202 

41 

20 

73  997 

81803 

18  197 

92  194 

40 

21 

74  017 

81  831 

18  169 

92  186 

39 

22 

74  036 

81  858 

18  142 

92  177 

38 

23 

74  055 

81  886 

18  114 

92  169 

37 

24 

74  074 

81913 

18  087 

92  161 

36 

25 

74  093 

81941 

18  059 

92  152 

35 

26 

74  113 

81968 

18  032 

92  144 

34 

27 

74  132 

81996 

18  004 

92  136 

33 

28 

74  151 

82  023 

17  977 

92  127 

32 

29 

74  170 

82  051 

17  949 

92  119 

31 

30 

74  189 

82  078 

17  922 

92  111 

30 

31 

74  208 

82  106 

17  894 

92  102 

29 

32 

74  227 

82  133 

17  867 

92  094 

28 

33 

74  246 

82  161 

17  839 

92  086 

27 

34 

74  265 

82  188 

17  812 

92  077 

26 

35 

74  284 

82  215 

17  785 

92  069 

25 

36 

74  303 

82  243 

17  757 

92  060 

24 

37 

74  322 

82  270 

17  730 

92  052 

23 

38 

74  341 

82  298 

17  702 

92  044 

22 

39 

74  360 

82  325 

17  675 

92  035 

21 

40 

74  379 

82  352 

17  648 

92  027 

20 

41 

74  398 

82  380 

17  620 

92  018 

19 

42 

74  417 

82  407 

17  593 

92  010 

18 

43 

74  436 

82  435 

17  565 

92  002 

17 

44 

74  455 

82  462 

17  538 

91  993 

16 

45 

74  474 

82  489 

17  511 

91985 

15 

46 

74  493 

82  517 

17  483 

91976 

14 

47 

74  512 

82  544 

17  456 

91  968 

13 

48 

74  531 

82  571 

17  429 

91959 

12 

49 

74  549 

82  599 

17  401 

91951 

11 

50 

74  568 

82  626 

17  374 

91942 

10 

51 

74  587 

82  653 

17  347 

91934 

9 

52 

74  606 

82  681 

17  319 

91925 

8 

53 

74  625 

82  708 

17  292 

91  917 

7 

54 

74  644 

82  735 

17  265 

91  908 

6 

55 

74  662 

82  762 

17  238 

91900 

5 

56 

74  681 

82  790 

17  210 

91  891 

4 

57 

74  700 

82  817 

17  183 

91883 

3 

58 

74  719 

82  844 

17  156 

91874 

2 

59 

74  737 

82  871 

17  129 

91  866 

1 

60 

74  756 

82  899 

Q 

17  101 
in 

91857 

— 9 

log  sin 

0 

r 

r 

y 
log  cos 

a 

log  cot 

1U 

log  tan 

34( 


t 

log  sin 

log  tan 

Q 

log  cot 

1  A 

log  cos 

f 

0 

74  756 

y 

82S99 

— iu — 

17  101 

91857 

60 

1 

74  775 

82  926 

17  074 

91  849 

59 

2 

74  794 

82  953 

17  047 

91  840 

58 

3 

74  812 

82  980 

17  020 

91  832 

57 

4 

74  831 

83  008 

16  992 

91823 

56 

5 

74  850 

83  035 

16  965 

91815 

55 

6 

74  868 

83  062 

16  938 

91  806 

54 

7 

74  887 

83  089 

16  911 

91  798 

53 

8 

74  906 

83  117 

16  883 

91  789 

52 

9 

74  924 

83  144 

16  856 

91  781 

51 

10 

74  943 

83  171 

16  829 

91  772 

50 

11 

74  961 

83  198 

16  802 

91  763 

49 

12 

74  980 

83  225 

16  775 

91  755 

48 

13 

74  999 

83  252 

16  748 

91  746 

47 

14 

75  017 

83  280 

16  720 

91  738 

46 

15 

75  036 

83  307 

16  693 

91  729 

45 

16 

75  054 

83  334 

16  666 

91  720 

44 

17 

75  073 

83  361 

16  639 

91  712 

43 

18 

75  091 

83  388 

16  612 

91  703 

42 

19 

75  110 

83  415 

16  585 

91  695 

41 

20 

75  128 

83  442 

16  558 

91686 

40 

21 

75  147 

83  470 

16  530 

91  677 

39 

22 

75  165 

83  497 

16  503 

91  669 

38 

23 

75  184 

83  524 

16  476 

91  660 

37 

24 

75  202 

83  551 

16  449 

91651 

36 

25 

75  221 

83  578 

16  422 

91  643 

35 

26 

75  239 

83  605 

16  395 

91  634 

34 

27 

75  258 

83  632 

16  368 

91  625 

33 

28 

75  276 

83  659 

16  341 

91617 

32 

29 

75  294 

83  686 

16  314 

91  60S 

31 

30 

75  313 

83  713 

16  287 

91  599 

30 

31 

75  331 

83  740 

16  260 

91  591 

29 

32 

75  350 

83  768 

16  232 

91  582 

28 

33 

75  368 

83  795 

16  205 

91  573 

27 

34 

75  386 

83  822 

16  178 

91  565 

26 

35 

75  405 

83  849 

16  151 

91  556 

25 

36 

75  423 

83  876 

16  124 

91  547 

24 

37 

75  441 

83  903 

16  097 

91  538 

23 

38 

75  459 

83  930 

16  070 

91  530 

22 

39 

75  478 

83  957 

16  043 

91521 

21 

40 

75  496 

83  984 

16  016 

91^12 

20 

41 

75  514 

84  011 

15  989 

91  504 

19 

42 

75  533 

84  038 

15  962 

91  495 

18 

43, 

75  551 

84  065 

15  935 

914S6 

17 

44 

75  569 

84  092 

15  908 

91477 

16 

45 

75  587 

84  119 

15  881 

91  469 

15 

46 

75  605 

84  146 

15  854 

91460 

14 

47 

75  624 

84  173 

15  827 

91  451 

13 

48 

75  642 

84  200 

15  800 

91  442 

12 

49 

75  660 

84  227 

15  773 

91  433 

11 

50 

75  678 

84  254 

15  746 

91  425 

10 

51 

75  696 

84  280 

15  720 

91  416 

9 

52 

75  714 

84  307 

15  693 

91407 

8 

53 

75  733 

84  334 

15  666 

91398 

7 

54 

75  751 

84  361 

15  639 

913S9 

6 

55 

75  769 

84  3S8 

15  612 

91  381 

5 

56 

75  787 

84  415 

15  585 

91372 

4 

57 

75  805 

84  442 

15  558 

91  363 

3 

58 

75  823 

84  469 

15  531 

91  354 

2 

59 

75  841 

84  496 

15  504 

91345 

1 

60 

75  859 

Q 

84  523 

Q 

15  477 
in 

913^ 

Q 

-6 

r 

— y — 

log  cos 

log  cot 

— iu 

log  tan 

y 

log  sin 

f 

56 


55 


35 


r 

log  sin 
9 

75  859 

log  tan 

9 

84  523 

log  cot 

— 10— 

15  477 

log  cos 

Q 

r 

0 

91336 

60 

1 

75  877 

84  550 

15  450 

91328 

59 

2 

75  895 

84  576 

15  424 

91319 

58 

3 

75  913 

84  603 

15  397 

91310 

57 

4 

75  931 

84  630 

15  370 

91301 

56 

5 

75  949 

84  657 

15  343 

91  292 

55 

6 

75  967 

S4  684 

15  316 

91283 

54 

7 

75  9S5 

S4  711 

15  289 

91274 

53 

8 

76  003 

84  738 

15  262 

91  266 

52 

9 

76  021 

84  764 

15  236 

91  257 

51 

10 

76  039 

S4  791 

15  209 

91  248 

50 

11 

76  057 

84  818 

15  182 

91  239 

49 

12 

76  075 

S4  845 

15  155 

91  230 

48 

13 

76  093 

84  872 

15  128 

91221 

47 

14 

76  111 

84  899 

15  101 

91  212 

46 

15 

76  129 

84  925 

15  075 

91  203 

45 

16 

76  146 

84  952 

15  048 

91  194 

44 

17 

76  164 

84  979 

15  021 

91  185 

43 

18 

76  182 

S5  006 

14  994 

91  176 

42 

19 

76  200 

85  033 

14  967 

91  167 

41 

20 

76  218 

85  059 

14  941 

91  158 

40 

21 

76  236 

85  0S6 

14  914 

91  149 

39 

22 

76  253 

85  113 

14  887 

91  141 

38 

23 

76  271 

85  140 

14  860 

91  132 

37 

24 

76  289 

85  166 

14  834 

91  123 

36 

25 

76  307 

85  193 

14  807 

91  114 

35 

26 

76  324 

85  220 

14  780 

91  105 

34 

27 

76  342 

85  247 

14  753 

91  096 

33 

28 

76  360 

85  273 

14  727 

91087 

32 

29 

76  37S 

85  300 

14  700 

91  078 

31 

30 

76  395 

85  327 

14  673 

91  069 

30 

31 

76  413 

85  354 

14  646 

91060 

29 

32 

76  431 

85  380 

14  620 

91051 

28 

33 

76  448 

85  407 

14  593 

91042 

27 

34 

76  466 

85  434 

14  566 

91  033 

26 

35 

76  484 

85  460 

14  540 

91023 

25 

36 

76  501 

85  487 

14  513 

91014 

24 

37 

76  519 

85  514 

14  486 

91005 

23 

38 

76  537 

85  540 

14  460 

90  996 

22 

39 

76  554 

85  567 

14  433 

90  987 

21 

40 

76  572 

85  594 

14  406 

90  978 

20 

41 

76  590 

85  620 

14  380 

90  969 

19 

42 

76  607 

85  647 

14  353 

90  960 

18 

43 

76  625 

85  674 

14  326 

90  951 

17 

44 

76  642 

85  700 

14  300 

90  942 

16 

45 

76  660 

85  727 

14  273 

90  933 

15 

46 

76  677 

85  754 

14  246 

90  924 

14 

47 

76  695 

85  780 

14  220 

90  915 

13 

48 

76  712 

85  807 

14  193 

90  906 

12 

49 

76  730 

85  834 

14  166 

90  896 

11 

50 

76  747 

85  860 

14  140 

90  887 

10 

51 

76  765 

85  887 

14  113 

90  878 

9 

52 

76  782 

85  913 

14  087 

90  869 

8 

53 

76SOO 

85  940 

14  060 

90  860 

7 

54 

76  817 

85  967 

14  033 

90  851 

6 

55 

76  835 

85  993 

14  007 

90  842 

5 

56 

76  852 

86  020 

13  980 

90  832 

4 

57 

76  870 

86  046 

13  954 

90  823 

3 

58 

76  887 

86  073 

13  927 

90  814 

2 

59 

76  904 

86  100 

13  900 

90  805 

1 

60 

76  922 

Q 

86  126 

9 

log  cot 

13  874 
—10— 

log  tan 

90  796 
n 

0 

t 

J7 

log  COS 

a 

log  sin 

r 

8 

.6° 

45 

r 

log  sin 

9 

76  922 

log  tan 

9 

86  126 

log  cot 

— 10— 

13  874 

log  cos 
g 

t 

0 

90  796 

60 

1 

76  939 

86  153 

13  847 

90  787 

59 

2 

76  957 

86  179 

13  821 

90  777 

58 

3 

76  974 

86  206 

13  794 

90  768 

57 

4 

76  991 

86  232 

13  768 

90  759 

56 

5 

77  009 

86  259 

13  741 

90  750 

55  • 

6 

77  026 

86  285 

13  715 

90  741 

54 

7 

77  043 

86  312 

13  688 

90  731 

53 

8 

77  061 

86  338 

13  662 

90  722 

52 

9 

77  078 

86  365 

13  635 

90  713 

51 

10 

77  095 

86  392 

13  608 

90  704 

50 

11 

77  112 

86  418 

13  582 

90  694 

49 

12 

77  130 

86  445 

13  555 

90  685 

48 

13 

77  147 

86  471 

13  529 

90  676 

47 

14 

77  164 

86  498 

13  502 

90  667 

46 

15 

77  181 

86  524 

13  476 

90  657 

45 

16 

77  199 

86  551 

13  449 

90  648 

44 

17 

77  216 

86  577 

13  423 

90  639 

43 

18 

77  233 

86  603 

13  397 

90  630 

42 

19 

77  250 

86  630 

13  370 

90  620 

41 

20 

77  268 

86  656 

13  344 

90  611 

40 

21 

77  285 

86  683 

13  317 

90  602 

39 

22 

77  302 

86  709 

13  291 

90  592 

38 

23 

77  319 

86  736 

13  264 

90  583 

37 

24 

77  336 

86  762 

13  238 

90  574 

36 

25 

77  353 

86  789 

13  211 

90  565 

35 

26 

77  370 

86  815 

13  185 

90  555 

34 

27 

77  387 

86  842 

13  158 

90  546 

33 

28 

77  405 

86  868 

13  132 

90  537 

32 

29 

77  422 

86  894 

13  106 

90  527 

31 

30 

77  439 

86  921 

13  079 

90  518 

30 

31 

77  456 

86  947 

13  053 

90  509 

29 

32 

77  473 

86  974 

13  026 

90  499 

28 

33 

77  490 

87  000 

13  000 

90  490 

27 

34 

77  507 

87  027 

12  973 

90  480 

26 

35 

77  524 

87  053 

12  947 

90  471 

25 

36 

77  541 

87  079 

12  921 

90  462 

24 

37 

77  558 

87  106 

12  894 

90  452 

23 

38 

77  575 

87  132 

12  868 

90  443 

22 

39 

77  592 

87  158 

12  842 

90  434 

21 

40 

77  609 

87  185 

12  815 

90  424 

20 

41 

77  626 

87  211 

12  789 

90  415 

19 

42 

77  643 

87  238 

12  762 

90  405 

18 

43 

77  660 

87  264 

12  736 

90  396 

17 

44 

77  677 

87  290 

12  710 

90  3S6 

16 

45 

77  694 

87  317 

12  683 

90  377 

15 

46 

77  711 

87  343 

12  657 

90  368 

14 

47 

77  728 

87  369 

12  631 

90  358 

13 

48 

77  744 

87  396 

12  604 

90  349 

12 

49 

77  761 

87  422 

12  578 

90  339 

11 

50 

77  778 

87  448 

12  552 

90  330 

10 

51 

77  795 

87  475 

12  525 

90  320 

9 

52 

77  812 

87  501 

12  499 

90  311 

k-8 

53 

77  829 

87  527 

12  473 

90  301 

7 

54 

77  846 

87  554 

12  446 

90  292 

6 

55 

77  862 

87  580 

12  420 

90  282 

5 

56 

77  879 

87  606 

12  394 

90  273 

4 

57 

77  896 

87  633 

12  367 

90  263 

3 

58 

77  913 

87  659 

12  341 

90  254 

2 

59 

77  930 

87  685 

12  315 

90  244 

1 

60 

77  946 

Q 

87  711 

— 9— 

log  cot 

12  289 

1Q 

90  235 

0 

0 

t 

a 

log  cos 

log  tan 

a 

log  sin 

t 

54 


53 


46 

37° 

t 

log  sin 

Q 

log  tan 

Q 

log  cot 

1  f\ 

log  cos 

Q 

t 

0 

77  946 

87  711 

— ±u — 

12  289 

90  235 

60 

1 

77  963 

87  738 

12  262 

90  225 

59 

2 

77  980 

87  764 

12  236 

90  216 

58 

3 

77  997 

87  790 

12  210 

90  206 

57 

4 

78  013 

87  817 

12  183 

90  197 

56 

5 

78  030 

87  843 

12  157 

90  187 

55 

6 

78  047 

87  869 

12  131 

90  178 

54 

7 

78  063 

87  895 

12  105 

90  168 

53 

8 

78  080 

87  922 

12  078 

90  159 

52 

9 

78  097 

87  948 

12  052 

90  149 

51 

10 

78  113 

87  974 

12  026 

90  139 

50 

11 

78  130 

88  000 

12  000 

90  130 

49 

12 

78  147 

88  027 

11973 

90  120 

48 

13 

78  163 

88  053 

11947 

90  111 

47 

14 

78  180 

88  079 

11921 

90  101 

46 

15 

78  197 

88  105 

11895 

90  091 

45 

16 

78  213 

88  131 

11869 

90  082 

44 

17 

78  230 

88  158 

11842 

90  072 

43 

18 

78  246 

88  184 

11816 

90  063 

42 

19 

78  263 

88  210 

11  790 

90  053 

41 

20 

78  280 

88  236 

11  764 

90  043 

40 

21 

78  296 

88  262 

11  738 

90  034 

39 

22 

78  313 

88  289 

11  711 

90  024 

38 

23 

78  329 

88  315 

11685 

90  014 

37 

24 

78  346 

88  341 

11659 

90  005 

36 

25 

78  362 

88  367 

11633 

89  995 

35 

26 

78  379 

88  393 

11607 

89  985 

34 

27 

78  395 

88  420 

11580 

89  976 

33 

28 

78  412 

88  446 

11554 

89  966 

32 

29 

78  428 

88  472 

11528 

89  956 

31 

30 

78  445 

88  498 

11502 

89  947 

30 

31 

78  461 

88  524 

11476 

89  937 

29 

32 

78  478 

88  550 

11450 

89  927 

28 

33 

78  494 

88  577 

11423 

89  918 

27 

34 

78  510 

88  603 

11397 

89  908 

26 

35 

78  527 

88  629 

11371 

89  898 

25 

36 

78  543 

88  655 

11345 

89  888 

24 

37 

78  560 

88  681 

11319 

89  879 

23 

38 

78  576 

88  707 

11293 

89  869 

22 

39 

78  592 

88  733 

11267 

89  859 

21 

40 

78  609 

88  759 

11241 

89  849 

20 

41 

78  625 

88  786 

11214 

89  840 

19 

42 

78  642 

88  812 

11  188 

89  830 

18 

43 

78  658 

88  838 

11  162 

89  820 

17 

44 

78  674 

88  864 

11  136 

89  810 

16 

45 

78  691 

88  890 

11  110 

89  801 

15 

46 

78  707 

88  916 

11084 

89  791 

14 

47 

78  723 

88  942 

11058 

89  781 

13 

48 

78  739 

88  968 

11032 

89  771 

12 

49 

78  756 

88  994 

11006 

89  761 

11 

50 

78  772 

89  020 

10  980 

89  752 

10 

51 

78  788 

89  046 

10  954 

89  742 

9 

52 

78  805 

89  073 

10  927 

89  732 

8 

53 

78  821 

89  099 

10  901 

89  722 

7 

54 

78  837 

89  125 

10  875 

89  712 

6 

55 

78  853 

89  151 

10  849 

89  702 

5 

56 

78  869 

89  177 

10  823 

89  693 

4 

57 

78  886 

89  203 

10  797 

89  683 

3 

58 

78  902 

89  229 

10  771 

89  673 

2 

59 

78  918 

89  255 

10  745 

89  663 

1 

60 

78  934 

89  281 

10  719 

89  653 

0 

n 

Q 

—10  — 

log  tan 

o 

t 

log  COS 

log  cot 

y 

log  sin 

t 

38( 


f 

log  sin 

9 

78  934 

log  tan 

9 

89  281 

log  cot 
—10— 
10  719 

log  cos 
g 

f 

0 

89  653 

60 

1 

78  950 

89  307 

10  693 

89  643 

59 

2 

78  967 

89  333 

10  667 

89  633 

58 

3 

78  983 

89  359 

10  641 

89  624 

57 

4 

78  999 

89  385 

10  615 

89  614 

56 

5 

79  015 

89  411 

10  589 

89  604 

55 

6 

79  031 

89  437 

10  563 

89  594 

54 

7 

79  047 

89  463 

10  537 

89  584 

53 

8 

79  063 

89  489 

10  511 

89  574 

52 

9 

79  079 

89  515 

10  485 

89  564 

51 

10 

79  095 

89  541 

10  459 

89  554 

50 

11 

79  111 

89  567 

10  433 

89  544 

49 

12 

79  128 

89  593 

10  407 

89  534 

48 

13 

79  144 

89  619 

10  381 

89  524 

47 

14 

79  160 

89  645 

10  355 

89  514 

46 

15 

79  176 

89  671 

10  329 

89  504 

45 

16 

79  192 

89  697 

10  303 

89  495 

44 

17 

79  208 

89  723 

10  277 

89  485 

43 

18 

79  224 

89  749 

10  251 

89  475 

42 

19 

79  240 

89  775 

10  225 

89  465 

41 

20 

79  256 

89  801 

10  199 

89  455 

40 

21 

79  272 

89  827 

10  173 

89  445 

39 

22 

79  288 

89  853 

10  147 

89  435 

38 

23 

79  304 

89  879 

10  121 

89  425 

37 

24 

79  319 

89  905 

10  095 

89  415 

36 

25 

79  335 

89  931 

10  069 

89  405 

35 

26 

79  351 

89  957 

10  043 

89  395 

34 

27 

79  367 

89  983 

10  017 

89  385 

33 

28 

79  383 

90  009 

09  991 

89  375 

32 

29 

79  399 

90  035 

09  965 

89  364 

31 

30 

79  415 

90  061 

09  939 

89  354 

30 

31 

79  431 

90  086 

09  914 

89  344 

29 

32 

79  447 

90  112 

09  888 

89  334 

28 

33 

79  463 

90  138 

09  862 

89  324 

27 

34 

79  478 

90  164 

09  836 

89  314 

26 

35 

79  494 

90  190 

09  810 

89  304 

25 

36 

79  510 

90  216 

09  784 

89  294 

24 

37 

79  526 

90  242 

09  758 

89  284 

23 

38 

79  542 

90  268 

09  732 

89  274 

22 

39 

79  558 

90  294 

09  706 

89  264 

21 

40 

79  573 

90  320 

09  680 

89  254 

20 

41 

79  589 

90  346 

09  654 

89  244 

19 

42 

79  605 

90  371 

09  629 

89  233 

18 

43 

79  621 

90  397 

09  603 

89  223 

17 

44 

79  636 

90  423 

09  577 

89  213 

16 

45 

79  652 

90  449 

09  551 

89  203 

15 

46 

79  668 

90  475 

09  525 

89  193 

14 

47 

79  684 

90  501 

09  499 

89  183 

13 

48 

79  699 

90  527 

09  473 

89  173 

12 

49 

79  715 

90  553 

09  447 

89  162 

11 

50 

79  731 

90  578 

09  422 

89  152 

10 

51 

79  746 

90  604 

09  396 

89  142 

9 

52 

79  762 

90  630 

09  370 

89  132 

8 

53 

79  778 

90  656 

09  344 

89  122 

7 

54 

79  793 

90  682 

09  318 

89  112 

6 

55 

79  809 

90  708 

09  292 

89  101 

5 

56 

79  825 

90  734 

09  266 

89  091 

4 

57 

79  840 

90  759 

09  241 

89  081 

3 

58 

79  856 

90  785 

09  215 

89  071 

2 

59 

79  872 

90  811 

09  189 

89  060 

1 

60 

79  887 

0 

90  837 

n 

09  163 

1  A 

89  050 

Q 

0 

f 

y 

log  cos 

y 

log  cot 

— xu — 

log  tan 

y 

log  sin 

f 

52 


51 


39 


r 

log  sin 
a 

log  tan 

Q 

log  cot 
10 

log  cos 
o 

f 

0 

79  887 

90  837 

09  163 

y 
89  050 

60 

1 

79  903 

90  863 

09  137 

89  040 

59 

2 

79  918 

90  889 

09  111 

89  030 

58 

3 

79  934 

90  914 

09  086 

89  020 

57 

4 

79  950 

90  940 

09  060 

89  009 

56 

5 

79  965 

90  966 

09  034 

88  999 

55 

6 

79  981 

90  992 

09  008 

88  989 

54 

7 

79  996 

91018 

08  982 

88  978 

53 

8 

80  012 

91043 

08  957 

88  968 

52 

9 

80  027 

91069 

08  931 

88  958 

51 

10 

80  043 

91095 

08  905 

88  948 

50 

11 

80  058 

91  121 

08  879 

88  937 

49 

12 

80  074 

91  147 

08  853 

88  927 

48 

13 

80  089 

91  172 

08  828 

88  917 

47 

14 

80  105 

91  198 

08  802 

88  906 

46 

15 

80  120 

91  224 

08  776 

88  896 

45 

16 

80  136 

91250 

08  750 

88  886 

44 

17 

80  151 

91276 

08  724 

88  875 

43 

18 

80  166 

91301 

08  699 

88  865 

42 

19 

80  182 

91  327 

08  673 

88  85£ 

41 

20 

80  197 

91353 

08  647 

88  844 

40 

21 

80  213 

91379 

08  621 

88  834 

39 

22 

80  228 

91404 

08  596 

88  824 

38 

23 

80  244 

91430 

08  570 

88  813 

37 

24 

.80  259 

91456 

08  544 

88  803 

36 

25 

80  274 

914S2 

08  518 

88  793 

35 

26 

80  290 

91  507 

08  493 

88  782 

34 

27 

80  305 

91533 

08  467 

88  772 

33 

28 

80  320 

91  559 

08  441 

88  761 

32 

29 

80  336 

915S5 

08  415 

88  751 

31 

30 

80  351 

91610 

08  390 

88  741 

30 

31 

80  366 

91636 

08  364 

88  730 

29 

32 

80  382 

91662 

08  338 

88  720 

28 

33 

80  397 

91688 

08  312 

88  709 

27 

34 

80  412 

91713 

08  287 

88  699 

26 

35 

80  428 

91  739 

08  261 

88  688 

25 

36 

80  443 

91  765 

08  235 

88  678 

24 

37 

80  458 

91  791 

08  209 

88  668 

23 

38 

80  473 

91816 

08  184 

88  657 

22 

39 

80.489 

91842 

08  158 

88  647 

21 

40 

80  504 

91868 

08  132 

88  636 

20 

41 

80  519 

91893 

08  107 

88  626 

19 

42 

80  534 

91919 

08  081 

88  615 

18 

43 

80  550 

91945 

08  055 

88  605 

17 

44 

80  565 

91971 

08  029 

88  594 

16 

45 

80  580 

91996 

08  004 

88  584 

15 

46 

80  595 

92  022 

07  978 

88  573 

14 

47 

80  610 

92  048 

07  952 

88  563 

13 

48 

80  625 

92  073 

07  927 

88  552 

12 

49 

80  641 

92  099 

07  901 

88  542 

11 

50 

80  656 

92  125 

07  875 

88  531 

10 

51 

80  671 

92  150 

07  850 

88  521 

9 

52 

80  686 

92  176 

07  824 

88  510 

8 

53 

80  701 

92  202 

07  798 

88  499 

7 

54 

80  716 

92  227 

07  773 

88  489 

6 

55 

80  731 

92  253 

07  747 

88  478 

5 

56 

80  746 

92  279 

07  721 

88  468 

4 

57 

80  762 

92  304 

07  696 

88  457 

3 

58 

80  777 

92  330 

07  670 

88  447 

2 

59 

80  792 

92  356 

07  644 

88  436 

1 

60 

80  807 

92  381 

07  619 

88  425 

0 

n 

o 

in 

o 

t 

y 

log  cos 

y 
log  cot 

log  tan 

log  sin 

f 

40° 

47 

t 

log  sin 
9 

log  tan 

9 

92  381 

log  cot 
— 10— 
07  619 

log  cos 

Q 

t 

0 

80  807 

88  425 

60 

1 

80  822 

92  407 

07  593 

88  415 

59 

2 

80  837 

92  433 

07  567 

88  404 

58 

3 

80  852 

92  458 

07  542 

88  3^4 

57 

4 

80  867 

92  484 

07  516 

88  383 

56 

5 

80  882 

92  510 

07  490 

88  372 

55 

6 

80  897 

92  535 

07  465 

88  362 

54 

7 

80  912 

92  561 

07  439 

88  351 

53 

8 

80  927 

92  587 

07  413 

88  340 

52 

9 

80  942 

92  612 

07  388 

88  330 

51 

10 

80  957 

92  638 

07  362 

88  319 

50 

11 

80  972 

92  663 

07  337 

88  308 

49 

12 

80  987 

92  689 

07  311 

88  298 

48 

13 

81002 

92  715 

07  285 

88  287 

47 

14 

81017 

92  740 

07  260 

88  276 

46 

15 

81032 

92  766 

07  234 

88  266 

45 

16 

81047 

92  792 

07  208 

88  255 

44 

17 

81061 

92  817 

07  183 

88  244 

43 

18 

81076 

92  843 

07  157 

88  234 

42 

19 

81091 

92  868 

07  132 

88  223 

41 

20 

81  106 

92  894 

07  106 

88  212 

40 

21 

81  121 

92  920 

07  080 

88  201 

39 

22 

81  136 

92  945 

07  055 

88  191 

38 

23 

81  151 

92  971 

07  029 

88  180 

37 

24 

81  166 

92  996 

07  004 

88  169 

36 

25 

81  180 

93  022 

06  978 

88  158 

35 

26 

81  195 

93  048 

06  952 

88  148 

34 

27 

81210 

93  073 

06  927 

88  137 

33 

28 

81225 

93  099 

06  901 

88  126 

32 

29 

81240 

93  124 

06  876 

88  115 

31 

30 

81254 

93  150 

06  850 

88  105 

30 

31 

81269 

93  175 

06  825 

88  094 

29 

32 

81284 

93  201 

06  799 

88  083 

28 

33 

81299 

93  227 

06  773 

88  072 

27 

34 

81314 

93  252 

06  748 

88  061 

26 

35 

81328 

93  278 

06  722 

88  051 

25 

36 

81343 

93  303 

06  697 

88  040 

24 

37 

81358 

93  329 

06  671 

88  029 

23 

38 

81372 

93  354 

06  646 

88  018 

22 

39 

81387 

93  380 

06  620 

88  007 

21 

40 

81402 

93  406 

06  594 

87  996 

20 

41 

81417 

93  431 

06  569 

87  985 

19 

42 

81431 

93  457 

06  543 

87  975 

18 

43 

81446 

93  482 

06  518 

87  964 

17 

44 

81461 

93  508 

06  492 

87  953 

16 

45 

81475 

93  533 

06  467 

87  942 

15 

46 

81490 

93  559 

06  441 

87  931 

14 

47 

81505 

93  584 

06  416 

87  920 

13 

48 

81  519 

93  610 

06  390 

87  909 

12 

49 

81534 

93  636 

06  364 

87  898 

11 

50 

81549 

93  661 

06  339 

87  887 

10 

51 

81563 

93  687 

06  313 

87  877 

9 

52 

81  578 

93  712 

06  288 

87  866 

8 

53 

81592 

93  738 

06  262 

87  855 

7 

54 

81607 

93  763 

06  237 

87  844 

6 

55 

81  622 

93  789 

06  211 

87  833 

5 

56 

81636 

93  814 

06  186 

87  822 

4 

57 

81651 

93  840 

06  160 

87  811 

3 

58 

81665 

93  865 

06  135 

87  800 

2 

59 

81680 

93  891 

06  109 

87  789 

1 

60 

81694 
— 9 — 

log  008 

93  916. 

— 9— 

log  cot 

06  084 
—10— 
log  tan 

87  778 

— 9 

log  sin 

0 

f 

^ 

50 


49 


48 


41 


42 


t 
0 

log  sin 

9 

81  694 

log  tan 

9 

93  916 

log  cot 
—10— 
06  084 

log  cos 
9 

87  778 

r 
60 

t 
0 

log  sin 

9 

82  551 

log  tan 

9 

95  444 

log  cot 

log  cos 

Q 

t 

04  556 

87  107 

60 

1 

81  709 

93  942 

06  058 

87  767 

59 

1 

82  565 

95  469 

04  531 

87  096 

59 

2 

81  723 

93  967 

06  033 

87  756 

58 

2 

82  579 

95  495 

04  505 

87  085 

58 

3 

81  738 

93  993 

06  007 

87  745 

57 

3 

82  593 

95  520 

04  480 

87  073 

57 

4 

81  752 

94  018 

05  982 

87  734 

56 

4 

82  607 

95  545 

04  455 

87  062 

56 

5 

81767 

94  044 

05  956 

87  723 

55 

5 

82  621 

95  571 

04  429 

87  050 

55 

6 

81  781 

94  069 

05  931 

87  712 

54 

6 

82  635 

95  596 

04  404 

87  039 

54 

7 

SI  796 

94  095 

05  905 

87  701 

53 

7 

82  649 

95  622 

04  378 

87  028 

53 

8 

81810 

94  120 

05  880 

87  690 

52 

8 

82  663 

95  647 

04  353 

87  016 

52 

9 

81  825 

94  146 

05  854 

87  679 

51 

9 

82  677 

95  672 

04  328 

87  005 

51 

10 

81839 

94  171 

05  829 

87  668 

50 

10 

82  691 

95  698 

04  302 

86  993 

50 

11 

81854 

94  197 

05  803 

87  657 

49 

11 

82  705 

95  723 

04  277 

86  982 

49 

12 

81868 

94  222 

05  778 

87  646 

48 

12 

82  719 

95  748 

04  252 

86  970 

48 

13 

81882 

94  248 

05  752 

87  635 

47 

13 

82  733 

95  774 

04  226 

86  959 

47 

14 

81897 

94  273 

05  727 

87  624 

46 

14 

82  747 

95  799 

04  201 

86  947 

46 

15 

81911 

94  299 

05  701 

87  613 

45 

15 

82  761 

95  825 

04  175 

86  936 

45 

16 

81926 

94  324 

05  676 

87  601 

44 

16 

82  775 

95  850 

04  150 

86  924 

44 

17 

81940 

94  350 

05  650 

87  590 

43 

17 

82  788 

95  875 

04  125 

86  913 

43 

18 

81  955 

94  375 

05  625 

87  579 

42 

18 

82  802 

95  901 

04  099 

86  902 

42 

19 

81969 

94  401 

05  599 

87  568 

41 

19 

82  816 

95  926 

04  074 

86  890 

41 

20 

81983 

94  426 

05  574 

87  557 

40 

20 

82  830 

95  952 

04  048 

86  879 

40 

21 

81998 

94  452 

05  548 

87  546 

39 

21 

82  844 

95  977 

04  023 

86  867 

39 

22 

82  012 

94  477 

05  523 

87  535 

38 

22 

82  858 

96  002 

03  998 

86  855 

38 

23 

82  026 

94  503 

05  497 

87  524 

37 

23 

82  872 

96  028 

03  972 

86  844 

37 

24 

82  041 

94  528 

05  472 

87  513 

36 

24 

82  885 

96  053 

03  947 

86  832 

36 

25 

82  055 

94  554 

05  446 

87  501 

35 

25 

82  899 

96  078 

03  922 

86  821 

35 

26 

82  069 

94  579 

05  421 

87  490 

34 

26 

82  913 

96  104 

03  896 

86  809 

34 

27 

82  084 

94  604 

05  396 

87  479 

33 

27 

82  927 

96  129 

03  871 

86  798 

33 

28 

82  098 

94  630 

05  370 

87  468 

32 

28 

82  941 

96  155 

03  845 

86  786 

32 

29 

82  112 

94  655 

05  345 

87  457 

31 

29 

82  955 

96  180 

03  820 

86  775 

31 

30 

82  126 

94  681 

05  319 

87  446 

30 

30 

82  968 

96  205 

03  795 

86  763 

30 

31 

82  141 

94  706 

05  294 

87  434 

29 

31 

82  982 

96  231 

03  769 

86  752 

29 

32 

82  155 

94  732 

05  268 

87  423 

28 

32 

82  996 

96  256 

03  744 

86  740 

28 

33 

82  169 

94  757 

05  243 

87  412 

27 

33 

83  010 

96  2S1 

03  719 

86  728 

27 

34 

82  184 

94  783 

05  217 

87  401 

26 

34 

83  023 

96  307 

03  693 

86  717 

26 

35 

82  198 

94  808 

05  192 

87  390 

25 

35 

83  037 

96  332 

03  668 

86  705 

25 

36 

82  212 

94  834 

05  166 

87  378 

24 

36 

83  051 

96  357 

03  643 

86  694 

24 

37 

82  226 

94  859 

05  141 

87  367 

23 

37 

83  065 

96  383 

03  617 

86  682 

23 

38 

82  240 

94  884 

05  116 

87  356 

22 

38 

83  078 

96  408 

03  592 

86  670 

22 

39 

82  255 

94  910 

05  090 

87  345 

21 

39 

83  092 

96  433 

03  567 

86  659 

21 

40 

82  269 

94  935 

05  065 

87  334 

20 

40 

83  106 

96  459 

03  541 

86  647 

20 

41 

82  283 

94  961 

05  039 

87  322 

19 

41 

83  120 

96  484 

03  516 

86  635 

19 

42 

82  297 

94  986 

05  014 

87  311 

18 

42 

83  133 

96  510 

03  490 

86  624 

18 

43 

82  311 

95  012 

04  988 

87  300 

17 

43 

83  147 

96  535 

03  465 

86  612 

17 

44 

82  326 

95  037 

04  963 

87  288 

16 

44 

83  161 

96  560 

03  440 

86  600 

16 

45 

82  340 

95  062 

04  938 

87  277 

15 

45 

83  174 

96  586 

03  414 

86  589 

15 

46 

82  354 

95  088 

04  912 

87  266 

14 

46 

83  188 

96  611 

03  389 

86  577 

14 

47 

82  368 

95  113 

04  887 

87  255 

13 

47 

83  202 

96  636 

03  364 

86  565 

13 

48 

82  382 

95  139 

04  861 

87  243 

12 

48 

83  215 

96  662 

03  338 

86  554 

12 

49 

82  396 

95  164 

04  836 

87  232 

11 

49 

83  229 

96  687 

03  313 

86  542 

11 

50 

82  410 

95  190 

04  810 

87  221 

10 

50 

83  242 

96  712 

03  2S8 

86  530 

10 

51 

82  424 

95  215 

04  785 

87  209 

9 

51 

83  256 

96  738 

03  262 

86  5 IS 

9 

52 

82  439 

95  240 

04  760 

87  198 

8 

52 

83  270 

96  763 

03  237 

86  507 

8 

53 

82  453 

95  266 

04  734 

87  187 

7 

53 

83  283 

96  788 

03  212 

86  495 

7 

54 

82  467 

95  291 

04  709 

87  175 

6 

54 

83  297 

96  814 

03  186 

86  483 

6 

55 

82  481 

95  317 

04  683 

87  164 

5 

55 

83  310 

96  839 

03  161 

86  472 

5 

56 

82  495 

95  342 

04  658 

87  153 

4 

56 

83  324 

96  864 

03  136 

86  460 

4 

57 

82  509 

95  368 

04  632 

87  141 

3 

57 

83  338 

96  890 

03  110 

86  448 

3 

58 

82  523 

95  393 

04  607 

87  130 

2 

58 

83  351 

96  915 

03  0S5 

86  436 

2 

59 

82  537 

95  418 

04  5S2 

87  119 

1 

59 

83  365 

96  940 

03  060 

S6  425 

1 

60 

82  551 
n 

95  444 

Q 

04  556 
in 

87  107 

Q 

0 

60 

83  378 

Q 

96  966 
g 

03  034 
in 

86  413 

0 

0 

f 

y 

log  cos 

y — 

log  cot 

log  tan 

a 

log  sin 

f 

f 

y 
log  cos 

log  cot 

log  tan 

j 
log  sin 

r 

48c 


47< 


43 


r 

log  sin 
n 

log  tan 

9 

96  966 

log  cot 
2.0 

log  cos 

f 

0 

83  378 

03  034 

86  413 

60 

1 

83  392 

96  991 

03  009 

86  401 

59 

2 

83  405 

97  016 

02  984 

86  389 

58 

3 

83  419 

97  042 

02  958 

86  377 

57 

4 

83  432 

97  067 

02  933 

86  366 

56 

5 

S3  446 

97  092 

02  908 

86  354 

55 

6 

83  459 

97  118 

02  882 

86  342 

54 

7 

83  473 

97  143 

02  857 

86  330 

53 

8 

83  486 

97  168 

02  832 

86  318 

52 

9 

83  500 

97  193 

02  807 

86  306 

51 

10 

83  513 

97  219 

02  781 

86  295 

50 

11 

83  527 

97  244 

02  756 

86  283 

49 

12 

83  540 

97  269 

02  731 

86  271 

48 

13 

83  554 

97  295 

02  705 

86  259 

47 

14 

83  567 

97  320 

02  680 

86  247 

46 

15 

83  581 

97  345 

02  655 

86  235 

45 

16 

83  594 

97  371 

02  629 

86  223 

44 

17 

83  608 

97  396 

02  604 

86  211 

43 

18 

83  621 

97  421 

02  579 

86  200 

42 

19 

83  634 

97  447 

02  553 

86  188 

41 

20 

83  648 

97  472 

02  528 

86  176 

40 

21 

83  661 

97  497 

02  503 

86  164 

39 

22 

83  674 

97  523 

02  477 

86  152 

38 

23 

83  688 

97  548 

02  452 

86  140 

37 

24 

83  701 

97  573 

02  427 

86  128 

36 

25 

83  715 

97  598 

02  402 

86  116 

35 

26 

83  728 

97  624 

02  376 

86  104 

34 

27 

83  741 

97  649 

02  351 

86  092 

33 

28 

83  755 

97  674 

02  326 

86  080 

32 

29 

83  768 

97  700 

02  300 

86  068 

31 

30 

83  781 

97  725 

02  275 

86  056 

30 

31 

83  795 

97  750 

02  250 

86  044 

29 

32 

83  808 

97  776 

02  224 

86  032 

28 

33 

83  821 

97  801 

02  199 

86  020 

27 

34 

83  834 

97  826 

02  174 

86  008 

26 

35 

83  848 

97  851 

02  149 

85  996 

25 

36 

83  861 

97  877 

02  123 

85  984 

24 

37 

83  874 

97  902 

02  098 

85  972 

23 

38 

83  887 

97  927 

02  073 

85  960 

22 

39 

83  901 

97  953 

02  047 

85  948 

21 

40 

83  914 

97  978 

02  022 

85  936 

20 

41 

83  927 

98  003 

01997 

85  924 

19 

42 

83  940 

98  029 

01971 

85  912 

18 

43 

83  954 

98  054 

01946 

85  900 

17 

44 

83  967 

98  079 

01921 

85  888 

16 

45 

83  980 

98104 

01896 

85  876 

15 

46 

83  993 

98  130 

01870 

85  864 

14 

47 

84  006 

98  155 

01845 

85  851 

13 

48 

84  020 

98  180 

01820 

85  839 

12 

49 

84  033 

98  206 

01  794 

85  827 

11 

50 

84  046 

98  231 

01  769 

85  815 

10 

51 

84  059 

98  256 

01  744 

85  803 

9 

52 

84  072 

98  281 

01719 

85  791 

8 

53 

84  085 

98  307 

01693 

85  779 

7 

54 

84  098 

98  332 

01668 

85  766 

6 

55 

84  112 

98  357 

01643 

85  754 

5 

56 

84  125 

98  383 

01617 

85  742 

4 

57 

84  138 

98  408 

01  592 

85  730 

3 

58 

84  151 

98  433 

01  567 

85  718 

2 

59 

84  164 

98  458 

01  542 

85  706 

1 

60 

84  177 
y 

98  484 

Q 

01  516 
in 

85  693 
n 

0 

r 

log  cos 

y 

log  cot 

— iu — 

log  tan 

y 

log  sin 

; 

44° 

49 

t 

log  sin 
g 

log  tan 

log  cot 
10 

log  cos 
n 

f 

0 

84  177 

y 
98  484 

01  516 

85  693 

60 

1 

84  190 

98  509 

01491 

85  681 

59 

2 

84  203 

98  534 

01466 

85  669 

58 

3 

84  216 

98  560 

01440 

85  657 

57 

4 

84  229 

98  585 

01415 

85  645 

56 

5 

84  242 

98  610 

01  390 

85  632 

55 

6 

84  255 

98  635 

01365 

85  620 

54 

7 

84  269 

98  661 

01339 

85  608 

53 

8 

84  282 

98  686 

01314 

85  596 

52 

9 

84  295 

98  711 

01289 

85  583 

51 

10 

84  308 

98  737 

01263 

85  571 

50 

11 

84  321 

98  762 

01238 

85  559 

49 

12 

84  334 

98  787 

01  213 

85  547 

48 

13 

84  347 

98  812 

01  188 

85  534 

47 

14 

84  360 

98  838 

01  162 

85  522 

46 

15 

84  373 

98  863 

01  137 

85  510 

45 

16 

84  385 

98  888 

01  112 

85  497 

44 

17 

84  398 

98  913 

01087 

85  485 

43 

18 

84  411 

98  939 

01061 

85  473 

42 

19 

84  424 

98  964 

01036 

85  460 

41 

20 

84  437 

98  989 

01011 

85  448 

40 

21 

84  450 

99  015 

00  985 

85  436 

39 

22 

84  463 

99  040 

00  960 

85  423 

38 

23 

84  476 

99  065 

00  935 

85  411 

37 

24 

84  489 

99  090 

00  910 

85  399 

36 

25 

84  502 

99  116 

00  884 

85  386 

35 

26 

84  515 

99  141 

00  859 

85  374 

34 

27 

84  528 

99  166 

00  834 

85  361 

33 

28 

84  540 

99  191 

00  809 

85  349 

32 

29 

84  553 

99  217 

00  783 

85  337 

31 

30 

84  566 

99  242 

00  758 

85  324 

30 

31 

84  579 

99  267 

00  733 

85  312 

29 

32 

84  592 

99  293 

00  707 

85  299 

28 

33 

84  605 

99  318 

00  682 

85  287 

27 

34 

84  618 

99  343 

00  657 

85  274 

26 

35 

84  630 

99  368 

00  632 

85  262 

25 

36 

84  643 

99  394 

00  606 

85  250 

24 

37 

84  656 

99  419 

00  581 

85  237 

23 

38 

84  669 

99  444 

00  556 

85  225 

22 

39 

84  682 

99  469 

00  531 

85  212 

21 

40 

84  694 

99  495 

00  505 

85  200 

20 

41 

84  707 

99  520 

00  480 

85  187 

19 

42 

84  720 

99  545 

00  455 

85  175 

18 

43 

84  733 

99  570 

00  430 

85  162 

17 

44 

84  745 

99  596 

00  404 

85  150 

16 

45 

84  758 

99  621 

00  379 

85  137 

15 

46 

84  771 

99  646 

00  354 

85  125 

14 

47 

84  784 

99  672 

00  328 

85  112 

13 

48 

84  796 

99  697 

00  303 

85  100 

12 

49 

84  809 

99  722 

00  278 

85  087 

11 

50 

84  822 

99  747 

00  253 

85  074 

10 

51 

84  835 

99  773 

00  227 

85  062 

9 

52 

84  847 

99  798 

00  202 

85  049 

8 

53 

84  860 

99  823 

00  177 

85  037 

7 

54 

84  873 

99  848 

00  152 

85  024 

6 

55 

84  885 

99  874 

00  126 

85  012 

5 

56 

84  898 

99  899 

00  101 

84  999 

4 

57 

84  911 

99  924 

00  076 

84  986 

3 

58 

84  923 

99  949 

00  051 

84  974 

2 

59 
60 

84  936 
84  949 

99  975 

00  025 
00  000 

84  961 
84  949 

1 
0 

00  000 

Q 

— 10 — 

log  cot 

—10— 

log  tan 

o 

f 

y 

log  cos 

log  sin 

f 

46 


45 


50 


TABLE  IV. 

For  Determining  with   Greater  Accuracy  than   can  be  done  by 

means  of    Table  III.  : 

1.    log  sin,  log  tan,  and  log  cot,  when  the  angle  is  between  0°  and  2°  ; 

2.    log  cos,  log  tan,  and  log  cot,  when  the  angle  is  between  88°  and  90°  ; 

3.    The  value  of  the  angle  when  the  logarithm  of  the  function  does  not 

lie  between  the  limits  8.  54  684  and  11.  45  316. 

FORMULAS  FOR  THE   USE    OF   THE  NUMBERS   S   AND   T. 

I.   When  the  angle  a  is  between  0°  and  2°  : 

log  sin  a  =  log  a"  +  S. 

log  a"  =  log  sin  a  —  &\ 

log  tan  a  =  log  a"  +  T. 

=  log  tan  a—  T, 

log  cot  a  =  colog  tan  a.                                             =  colog  cot  a  —  T. 

II.    When  the  angle  a  is  between  88°  and  90°  : 

log  COS  a  =  log  (90° -a)"  +  S. 

log  (90°-a)"  =  log  cos  a-S, 

log  cot  a  =  log  (90°— a)"  +  T. 

=  log  cot  a—  T, 

log  tan  a  =  colog  cot  a. 

=  colog  tan  a—T, 

and  a  =  90° -(90° -a). 

*oX«oo 

Values  of  S  a^d  T, 

a" 

S 

log  sin  a 

a" 

T 

log  tan  a 

a 

T 

log  tan  a 

0 

■ 

0 



5146 

8.  39  713 

4.  68  557 

4.68  557 

4.  68  567 

2  409 

4.68  556 

8.06  740 

200 

4.  68  558 

6. 98  660 

5  424 

4.68  568 

8.41999 

3  417 

4.68  555 

8.  21  920 

1726 

4.  68  559 

7.  92  263 

5  689 

4. 68  569 

8.  44  072 

3  823 

4.  68  555 

8.  26  795 

2  432 

4.  68  560 

8.  07  156 

5  941 

4.  68  570 

8.  45  955 

4190 

4. 68  554 

8. 30  776 

2  976 

4.  68  561 

8. 15  924 

6184 

4.  68  571 

8. 47  697 

4  840 

4.68  553 

8. 37  038 

3  434 

4.  68  562 

8.  22  142 

6417 

4.68  572 

8.49  305 

5  414 

4.  68  552 

8. 41  904 

3  838 

4.  68  563 

8.26  973 

6642 

4. 68  573 

8.  50  802 

5  932 

4. 68  551 

8. 45  872 

4  204 

4.  68  564 

8.  30  930 

6  859 

4. 68  574 

8.  52  200 

6  408 

4. 68  550 

8. 49  223 

4  540 

4.  68  565 

8.  34  270 

7070 

4. 68  575 

8.  53  516 

6  633 

4.  68  550 

8.  50  721 

4  699 

4.  68  565 

8. 35  766 

7173 

4. 68  575 

8.  54  145 

6  851 

4. 68  549 

8.  52  125 

4  853 

4.  68  566 

8.  37  167 

7  274 

8.  54  753 

7  267 

8.54  684 

5  146 

8. 39  713 

a" 

S 

log  sin  a 

a". 

T 

log  tan  a 

a 

T 

log  tan  a 

TABLE  V.— Circumferences  and  Areas  of  Circles,  si 

If  N=  the  radius  of  the  circle,  the  circumference  =  2 

irN. 

If  2f=  the  radius  of  the  circle,  the  area 

=  irN2. 

If  JV=  the  circumference  of  the  circle,  the  radius  =  —■#". 

*7T 

If  X=  the  circumfereuce  of  the  circle,  the  area     =  —  iV2. 

47T 

N 

lirN 

TT.V2 

Lit 

2tt 

4* 

I 

50 

2irtf 

TriV™ 

J-2VT 
2tt 

4tt 

0 

0.00 

0.0 

0.000 

0.00 

314. 16 

7  854 

7.96 

198.  94 

1 

6.28 

3.1 

0.159 

0.08 

51 

320. 44 

8171 

8.12 

206.  98 

2 

12.57 

12.6 

0.318 

0.32 

52 

326.  73 

8  495 

8.28 

215. 18 

3 

18.85 

28.3 

0.477 

0.72 

53 

333. 01 

8  825 

8.44 

223. 53 

4 

25.13 

50.3 

0.637 

1.27 

54 

339.  29 

9161 

8.59 

232. 05 

5 

31.42 

78.5 

0.796 

1.99 

55 

345.  58 

9  503 

8.75 

240.  72 

6 

37.70 

113.1 

0.955 

2.86 

56 

351.  86 

9  852 

8.91 

249.  55 

7 

43.98 

153.9 

1.114 

3.90 

57 

358. 14 

10  207 

9.07 

258.  55 

8 

50.27 

201.1 

1.273 

5.09 

58 

364.42 

10  568 

9.23 

267.  70 

9 

56.55 

254.5 

1.432 

6.45 

59 

370.  71 

10  936 

9.39 

277. 01 

10 

62.83 

314.2 

1.  592 

7.96 

60 

376.  99 

11310 

9.55 

286. 48 

11 

69.12 

380.1 

1.751 

9.63 

61 

383.  27 

11690 

9.71 

296.  11 

12 

75.40 

452.4 

1.910 

11.46 

62 

389.  56 

12  076 

9.87 

305.  90 

13 

81.68 

530.9 

2.069 

13.45 

63 

395.  84 

12  469 

10.03 

315.  84 

14 

87.96 

615.8 

2.228 

15.60 

64 

402. 12 

12  868 

10.19 

325.95 

15 

94.25 

706.9 

2.387 

17.90 

65 

408. 41 

13  273 

10.35 

336.  21 

16 

100.  53 

804.2 

2.546 

20.37 

66 

414.  69 

13  685 

10.50 

346.  64 

17 

106.81 

907.9 

2.706 

23.00 

67 

420. 97 

14103 

10.66 

357.  22 

18 

113. 10 

1017.9 

2.865 

25.78 

68 

427.  26 

14  527 

10.82 

367.  97 

19 

119.  38 

1  134. 1 

3.024 

28.73 

69 

433. 54 

14  957 

10.98 

378.  87 

20 

125.66 

1  256.  6 

3.183 

31.83 

70 

439.  82 

15  394 

11.14 

389.  93 

21 

131.95 

1  385. 4 

3.342 

35.09 

71 

446. 11 

15  837 

11.30 

401. 15 

22 

138.  23 

1  520.  5 

3.501 

38.52 

72 

452.  39 

16  286 

11.46 

412.  53 

23 

144.51 

1661.9 

3.661 

42.10 

73 

458.  67 

16  742 

11.62 

424. 07 

24 

150.  80 

1  809. 6 

3.820 

45.84 

74 

464.96 

17  203 

11.78 

435. 77 

25 

157.  08 

1  963. 5 

3.979 

49.74 

75 

471.  24 

17  671 

11.94 

447.  62 

26 

163.  36 

2 123. 7 

4.138 

53.79 

76 

477.  52 

18  H6 

12.10 

459.64 

27 

169.  65 

2  290.  2 

4.297 

58.01 

77 

483.  81 

18  627 

12.25 

471.  81 

28 

175.  93 

2  463.0 

4.456 

62.39 

78 

490.09 

19113 

12.41 

484. 15 

29 

182.  21 

2  642.1 

4.615 

66.92 

79 

496.  37 

19  607 

12.57 

496.64 

30 

188. 50 

2  827. 4 

4.775 

71.62 

80 

502.  65 

20106 

12.73 

509.  30 

31 

194. 78 

3  019. 1 

4.934 

76.47 

81 

508. 94 

20  612 

12.89 

522. 11 

32 

201.06 

3  217.0 

5.093 

81.49 

82 

515.22 

21124 

13.05 

535.  08 

33 

207.  35 

3  421.2 

5.252 

86.66 

83 

521.  50 

21642 

13.21 

548. 21 

34 

213.  63 

3  631.  7 

5.411 

91.99 

84 

527.  79 

22167 

13.37 

561.  50 

35 

219. 91 

3  848.  5 

5.570 

97.48 

85 

534.  07 

22  698 

13.53 

574.  95 

36 

226. 19 

4  071.5 

5.730 

103. 13 

86 

540.  35 

23  235 

13.69 

588.  55 

37 

232. 48 

4  300.8 

5.889 

108.  94 

87 

546.  64 

23  779 

13.85 

602.32 

38 

238. 76 

4  536.  5 

6.048 

114.91 

88 

552.  92 

24  328 

14.01 

616.  25 

39 

245.04 

4  778.4 

6.207 

121.04 

89 

559.  20 

24  885 

14.16 

630. 33 

40 

251.33 

5  026.  5 

6.366 

127.32 

90 

565.49 

25  447 

14.32 

644.58 

41 

257.  61 

5  281.0 

6.525 

133. 77 

91 

571.  77 

26  016 

14.48 

658.  98 

42 

263.  89 

5  541.8 

6.685 

140.  37 

92 

578.  05 

26  590 

14.64 

673.  54 

43 

270. 18 

5  808.  8 

6.844 

147. 14 

93 

584.  34 

27172 

14.80 

688.  27 

44 

276. 46 

6  082. 1 

7.003 

154.06 

94 

590.62 

27  759 

14.96 

703. 15 

45 

282.  74 

6  361.7 

7.162 

161. 14 

95 

596. 90 

28  353 

15.12 

718. 19 

46 

289. 03 

6  647.6 

7.321 

168.  39 

96 

603. 19 

28  953 

15.28 

733.  39 

47 

295.  31 

6  939.  8 

7.480 

175. 79 

97 

609.47 

29  559 

15.44 

748.  74 

48 

301.  59 

7  238.  2 

7.639 

183. 35 

98 

615.  75 

30  172 

15.60 

764.26 

49 

307.  88 

7  543. 0 

7.799 

191. 07 

99 

622.04 

30  791 

15.76 

779. 94 

50 

314. 16 

7  854. 0 

7.958 

198.94 

100 

628.  32 

31416 

15.92 

795.  77 

N 

2  7T.V 

TT.V2 

2tt 

4ir 

lirN 

ttN* 

2tt 

4tt 

52 


TABLE  VI. -NATURAL  FUNCTIONS. 


/ 

o° 

1° 

2° 

sin   cos 

3 

° 

4 

o 

r 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

COS 

o 

0000 

1000 

0175 

9998 

0349 

9994 

0523 

9986 

0698 

9976 

60 

1 

0003 

1000 

0177 

9998 

0352 

9994 

0526 

9986 

0700 

9975 

59 

2 

0006 

1000 

0180 

9998 

0355 

9994 

0529 

9986 

0703 

9975 

58 

3 

0009 

1000 

0183 

9998 

0358 

9994 

0532 

9986 

0706 

9975 

57 

4 

0012 

1000 

0186 

9998 

0361 

9993 

0535 

9986 

0709 

9975 

56 

5 

0015 

1000 

0189 

9998 

0364 

9993 

0538 

9986 

0712 

9975 

55 

6 

0017 

1000 

0192 

9998 

0366 

9993 

0541 

9985 

0715 

9974 

54 

7 

0020 

1000 

0195 

9998 

0369 

9993 

0544 

9985 

0718 

9974 

53 

8 

0023 

1000 

0198 

9998 

0372 

9993 

0547 

9985 

0721 

9974 

52 

9 

0026 

1000 

0201 

9998 

0375 

9993 

0550 

9985 

0724 

9974 

51 

io 

0029 

1000 

0204 

9998 

0378 

9993 

0552 

9985 

0727 

9974 

SO 

11 

0032 

1000 

0207 

9998 

0381 

9993 

0555 

9985 

0729 

9973 

49 

12 

0035 

1000 

0209 

9998 

0384 

9993 

0558 

9984 

0732 

9973 

48 

13 

0038 

1000 

0212 

9998 

0387 

9993 

0561 

9984 

0735 

9973 

47 

14 

0041 

1000 

0215 

9998 

0390 

9992 

0564 

9984 

0738 

9973 

46 

15 

0044 

1000 

0218 

9998 

0393 

9992 

0567 

9984 

0741 

9973 

45 

16 

0047 

1000 

0221 

9998 

0396 

9992 

0570 

9984 

0744 

9972 

44 

17 

0049 

1000 

0224 

9997 

0398 

9992 

0573 

9984 

0747 

9972 

43 

18 

0052 

1000 

0227 

9997 

0401 

9992 

0576 

9983 

0750 

9972 

42 

19 

0055 

1000 

0230 

9997 

0404 

9992 

0579 

9983 

0753 

9972 

41 

20 

0058 

1000 

0233 

9997 

0407 

9992 

0581 

9983 

0756 

9971 

40 

21 

0061 

1000 

0236 

9997 

0410 

9992 

0584 

9983 

0758 

9971 

39 

22 

0064 

1000 

0239 

9997 

0413 

9991 

0587 

9983 

0761 

9971 

38 

23 

0067 

1000 

0241 

9997 

0416 

9991 

0590 

9983 

0764 

9971 

37 

24 

0070 

1000 

0244 

9997 

0419 

9991 

0593 

9982 

0767 

9971 

36 

25 

0073 

1000 

0247 

9997 

0422 

9991 

0596 

9982 

0770 

9970 

35 

26 

0076 

1000 

0250 

9997 

0425 

9991 

0599 

9982 

0773 

9970 

34 

27 

0079 

1000 

0253 

9997 

0427 

9991 

0602 

9982 

0776 

9970 

33 

28 

0081 

1000 

0256 

9997 

0430 

9991 

0605 

9982 

0779 

9970 

32 

29 

0084 

1000 

0259 

9997 

0433 

9991 

0608 

9982 

0782 

9969 

31 

30 

0087 

1000 

0262 

9997 

0436 

9990 

0610 

9981 

0785 

9969 

30 

31 

0090 

1000 

0265 

9996 

0439 

9990 

0613 

9981 

0787 

9969 

29 

32 

0093 

1000 

0268 

9996 

0442 

9990 

0616 

9981 

0790 

9969 

28 

33 

0096 

1000 

0270 

9996 

0445 

9990 

0619 

9981 

0793 

9968 

27 

34 

0099 

1000 

0273 

9996 

0448 

9990 

0622 

9981 

0796 

9968 

26 

35 

0102 

9999 

0276 

9996 

0451 

9990 

0625 

9980 

0799 

9968 

25 

36 

0105 

9999 

0279 

9996 

0454 

9990 

0628 

9980 

0802 

9968 

24 

37 

0108 

9999 

0282 

9996 

0457 

9990 

0631 

9980 

0805 

9968 

23 

38 

0111 

9999 

0285 

9996 

0459 

9989 

0634 

9980 

0808 

9967 

22 

39 

0113 

9999 

0288 

9996 

0462 

9989 

0637 

9980 

0811 

9967 

21 

40 

0116 

9999 

0291 

9996 

0465 

9989 

0640 

9980 

0814 

9967 

20 

41 

0119 

9999 

0294 

9996 

0468 

9989 

0642 

9979 

0816 

9967 

19 

42 

0122 

9999 

0297 

9996 

0471 

9989 

0645 

9979 

0819 

9966 

18 

43 

0125 

9999 

0300 

9996 

0474 

9989 

0648 

9979 

0822 

9966 

17 

44 

0128 

9999 

0302 

9995 

0477 

9989 

0651 

9979 

0825 

9966 

16 

45 

0131 

9999 

0305 

9995 

0480 

9988 

0654 

9979 

0828 

9966 

15 

46 

0134 

9999 

0308 

9995 

0483 

9988 

0657 

9978 

0831 

9965 

14 

47 

0137 

9999 

0311 

9995 

0486 

9988 

0660 

9978 

0834 

9965 

13 

48 

0140 

9999 

0314 

9995 

0488 

9988 

0663 

9978 

0837 

9965 

12 

49 

0143 

9999 

0317 

9995 

0491 

9988 

0666 

9978 

0840 

9965 

11 

SO 

0145 

9999 

0320 

9995 

0494 

9988 

0669 

9978 

0843 

9964 

IO 

51 

0148 

9999 

0323 

9995 

0497 

9988 

0671 

9977 

0845 

9964 

9 

52 

0151 

9999 

0326 

9995 

QSfla 

9987 

0674 

9977 

0848 

9964 

8 

53 

0154 

9999 

0329 

9995 

0503 

9987 

0677 

9977 

0851 

9964 

7 

54 

0157 

9999 

0332 

9995 

0506 

9987 

0680-9977 

0854 

9963 

6 

55 

0160 

9999 

0334 

9994 

0509 

9987 

0683 

9977 

0857 

9963 

5 

56 

0163 

9999 

0337- 

-9994 

0512 

9987 

0686 

9976 

0860 

9963 

4 

57 

0166 

9999 

0340 

9994 

0515 

9987 

0689 

9976 

0863 

9963 

3 

58 

0169 

9999 

0343 

9994 

0518 

9987 

0692 

9976 

0866 

9962 

2 

59 

0172 

9999 

0346 

9994 

0520 

9986 

0695 

9976 

0869 

9962 

1 

60 

0175 

9999 

0349 

9994 

0523 

9986 

0698 

9976 

0872 

9962 

o 

cos   sin 

89° 

cos   sin 
88° 

cos   sin 

87° 

cos   sin 

86° 

cos 

sin 



r 

85° 

'1 

NATURAL 

SINES  AND 

COSINES. 

53 

r 

5° 

6° 

7° 

8° 

9° 

r 

sin   cog 

sin 

cos 

sin   cos 

sin   cos 

sin   cos 

O 

0872  9962 

1045 

9945 

1219  9925 

1392  9903 

1564  9877 

60 

1 

0874  9962 

1048 

9945 

1222  9925 

1395  9902 

1567  9876 

59 

2 

0877  9461 

1051 

9945 

1224  9925 

1397  9902 

1570  9876 

58 

3 

0880  9961 

1054 

9944 

1227  9924 

1400  9901 

1573  9876 

57 

4 

0883  9961 

1057 

9944 

1230  9924 

1403  9901 

1576  9875 

56 

5 

0886  9961 

1060 

9944 

1233  9924 

1406  9901 

1579  9875 

55 

6 

0889  9960 

1063 

9943 

1236  9923 

1409  9900 

1582  9874 

54 

7 

0892  9960 

1066 

9943 

1239  9923 

1412  9900 

1584  9874 

53 

8 

0895  9960 

1068 

9943 

1241  9923 

1415  9899 

1587  9873 

52 

9 

0898  9960 

1071 

9942 

1245  9922 

1418  9899 

1590  9873 

51 

io 

0901  9959 

1074 

9942 

1248  9922 

1421  9899 

1593  9872 

50 

11 

0903  9959 

1077 

9942 

1250  9922 

1423  9898 

1596  9872 

49 

12 

0906  9959 

1080 

9942 

1253  9921 

1426  9898 

1599  9871 

48 

13 

0909  9959 

1083 

9941 

1256  9921 

1429  9897 

1602  9871 

47 

14 

0912  9958 

1086 

9941 

1259  9920 

1432  9897 

1605  9870 

46 

15 

0915  9958 

1089 

9941 

1262  9920 

1435  9897 

1607  9870 

45 

]6 

0918  9958 

1092 

9940 

1265  9920 

1438  9896 

1610  9869 

44 

17 

0921  9958 

1094 

9940 

1268  9919 

1441  9896 

1613  9869 

43 

18 

0924  9957 

1097 

9940 

1271  9919 

1444  9895 

1616  9869 

42 

19 

0927  9957 

1100 

9939 

1274  9919 

1446  9895 

1619  9868 

41 

20 

0929  9957 

1103 

9939 

1276  9918 

1449  9894 

1622  9868 

40 

21 

0932  9956 

1106 

9939 

1279  9918 

1452  9894 

1625  9867 

39 

22 

0935  9956 

1109 

9938 

1282  9917 

1455  9894 

1628  9867 

38 

23 

0938  9956 

1112 

9938 

1285  9917 

1458  9893 

1630  9866 

37 

24 

0941  9956 

1115 

9938 

1288  9917 

1461  9893 

1633  9866 

36 

25 

0944  9955 

1118 

9937 

1291  9916 

1464  9892 

1636  9865 

35 

26 

0947  9955 

1120 

9937 

1294  9916 

1467  9892 

1639  9865 

34 

27 

0950  9955 

1123 

9937 

1297  9916 

1469  9891 

1642  9864 

33 

28 

0953  9955 

1126 

9936 

1299  9915 

1472  9891 

1645  9864 

32 

29 

0956  9954 

1129 

9936 

1302  9915 

1475  9891 

1648  9863 

31 

30 

0958  9954 

1132 

9936 

1305  9914 

1478  9890 

1650  9863 

30 

31 

0961  9954 

1135 

9935 

1308  9914 

1481  9890 

1653  9862 

29 

32 

9964  9953 

1138 

9935 

1311  9914 

1484  9889 

1656  9862 

28 

33 

0967  9953 

1141 

9935 

1314  9913 

1487  9889 

1659  9861 

27 

34 

0970  9953 

1144 

9934 

1317  9913 

1490  9888 

1662  9861 

26 

35 

0973  9553 

1146 

9934 

1320  9913 

1492  9888 

1665  9860 

25 

36 

0976  9952 

1149 

9934 

1323  9912 

1495  9888 

1668  9860 

24 

37 

0979  9952 

1152 

9933 

1325  9912 

1498  9887 

1671  9859 

23 

38 

0982  9952 

1155 

9933 

1328  9911 

1501  9887 

1673  9859 

22 

39 

0985  9951 

1158 

9933 

1331  9911 

1504  9886 

1676  9859 

21 

40 

0987  9951 

1161 

9932 

1334  9911 

1507  9886 

1679  9858 

20 

41 

0990  9951 

1164 

9932 

1337  9910 

1510  9885 

1682  9858 

19 

42 

0993  9951 

1167 

9932 

1340  9910 

1513  9885 

1685  9857 

18 

43 

0996  9950 

1170 

9931 

1343  9909 

1515  9884 

1688  9857 

17 

44 

0999  9950 

1172 

9931 

1346  9909 

1518  9884 

1691  9856 

16 

45 

1002  9950 

1175 

9931 

1349  9909 

1521  9884 

1693  9856 

15 

46 

1005  9949 

1178 

9930 

1351  9908 

1524  9883 

1696  9855 

14 

47 

1008  9949 

1181 

9930 

1354  9908 

1527  9883 

1699  9855 

13 

48 

1011  9949 

1184 

9930 

1357  9907 

1530  9882 

1702  9854 

12 

49 

1013  9949 

1187 

9929 

1360  9907 

1533  9882 

1705  9854 

11 

50 

1016  9948 

1190 

9929 

1363  9907 

1536  9881 

1708  9853 

IO 

51 

1019  9948 

1193 

9929 

1366  9906 

1538  9881 

1711  9853 

9 

52 

1022  9948 

1196 

9928 

1369  9906 

1541  9880 

1714  9852 

8 

53 

1025  9947 

1198 

9928 

1372  9905 

1544  9880 

1716  9852 

7 

54 

1028  9947 

1201 

9928 

1374  9905 

1547  9880 

1719  9851 

6 

55 

1031  9947 

1204 

9927 

1377  9905 

1550  9879 

1722  9851 

5 

56 

1034  9946 

1207 

9927 

1380  9904 

1553  9879 

1725  9850 

4 

57 

1037  9946 

1210 

9927 

1383  9904 

1556  9878 

1728  9850 

3 

58 

1039  9946 

1213 

9926 

1386  9903 

1559  9878 

1731  9849 

2 

59 

1042  9946 

1216 

9926 

1389  9903 

1561  9877 

1734  9849 

1 

60 

1045  9945 

1219 

9925 

1392  9903 

1564  9877 

1736  9848 

O 

cos    sin 

84° 

cos   sin 

83° 

cor    sin 

82° 

cos    Bin 

81° 

cos   sin 

80° 

f 

t 

54 

NATURAL 

SINES  AND 

COSINES. 

t 

io° 

11° 

12° 

13° 

14° 

t 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

o 

1736 

9848 

1908 

9816 

2079 

9781 

2250 

9744 

2419 

9703 

60 

1 

1739 

9848 

1911 

9816 

2082 

9781 

2252 

9743 

2422 

9702 

59 

2 

1742 

9847 

1914 

9815 

2085 

9780 

2255 

9742 

2425 

9702 

58 

3 

1745 

9847 

1917 

9815 

2088 

9780 

2258 

9742 

2428 

9701 

57 

4 

1748 

9846 

1920 

9814 

2090 

9779 

2261 

9741 

2431 

9700 

56 

5 

1751 

9846 

1922 

9813 

2093 

9778 

2264 

9740 

2433 

9699 

55 

6 

1754 

9845 

1925 

9813 

2096 

9778 

2267 

9740 

2436 

9699 

54 

7 

1757 

9845 

1928 

9812 

2099 

9777 

2269 

9739 

2439 

9698 

53 

8 

1759 

9844 

1931 

9812 

2102 

9777 

2272 

9738 

2442 

9697 

52 

9 

1762 

9843 

1934 

9811 

2105 

9776 

2275 

9738 

2445 

9697 

51 

10 

1765 

9843 

1937 

9811 

2108 

9775 

2278 

9737 

2447 

9696 

50 

11 

1768 

9842 

1939 

9810 

2110 

9775 

2281 

9736 

2450 

9695 

49 

12 

1771 

9842 

1942 

9810 

2113 

9774 

2284 

9736 

2453 

9694 

48 

13 

1774 

9841 

1945 

9809 

2116 

9774 

2286 

9735 

2456 

9694 

47 

14 

1777 

9841 

1948 

9808 

2119 

9773 

2289 

9734 

2459 

9693 

46 

15 

1779 

9840 

1951 

9808 

2122 

9772 

2292 

9734 

2462 

9692 

45 

16 

1782 

9840 

1954 

9807 

2125 

9772 

2295 

9733 

2464 

9692 

44 

17 

1785 

9839 

1957 

9807 

2127 

9771 

2298 

9732 

2467 

9691 

43 

18 

1788 

9839 

1959 

9806 

2130 

9770 

2300 

9732 

2470 

9690 

42 

19 

1791 

9838 

1962 

9806 

2133 

9770 

2303 

9731 

2473 

9689 

41 

20 

1794 

9838 

1965 

9805 

2136 

9769 

2306 

9730 

2476 

9689 

40 

21 

1797 

9837 

1968 

9804 

2139 

9769 

2309 

9730 

2478 

9688 

39 

22 

1799 

9837 

1971 

9804 

2142 

9768 

2312 

9729 

2481 

9687 

38 

23 

1802 

9836 

1974 

9803 

2145 

9767 

2315 

9728 

2484 

9687 

37 

24 

1805 

9836 

1977 

9803 

2147 

9767 

2317 

9728 

2487 

9686 

36 

25 

1808 

9835 

1979 

9802 

2150 

9766 

2320 

9727 

2490 

9685 

35 

26 

1811 

9835 

1982 

9802 

2153 

9765 

2323 

9726 

2493 

9684 

34 

27 

1814 

9834 

1985 

9801 

2156 

9765 

2326 

9726 

2495 

9684 

33 

28 

1817 

9834 

1988 

9800 

2159 

9764 

2329 

9725 

2498 

9683 

32 

29 

1819 

9833 

1991 

9800 

2162 

9764 

2332 

9724 

2501 

9682 

31 

30 

1822 

9833 

1994 

9799 

2164 

9763 

2334 

9724 

2504 

9681 

30 

31 

1825 

9832 

1997 

9799 

2167 

9762 

2337 

9723 

2507 

9681 

29 

32 

1828 

9831 

1999 

9798 

2170 

9762 

2340 

9722 

2509 

9680 

28 

33 

1831 

9831 

2002 

9798 

2173 

9761 

3343 

9722 

2512 

9679 

27 

34 

1834 

9830 

2005 

9797 

2176 

9760 

2346 

9721 

2515 

9679 

26 

35 

1837 

9830 

2008 

9796 

2179 

9760 

2349 

9720 

2518 

9678 

25 

36 

1S40 

9829 

2011 

9796 

2181 

9759 

2351 

9720 

2521 

9677 

24 

37 

1842 

9829 

2014 

9795 

2184 

9759 

2354 

9719 

2524 

9676 

23 

38 

1845 

9828 

2016 

9795 

2187 

9758 

2357 

9718 

2526 

9676 

22 

39 

1848 

9828 

2019 

9794 

2190 

9757 

2360 

9718 

2529 

9675 

21 

40 

1851 

9827 

2022 

9793 

2193 

9757 

2363 

9717 

2532 

9674 

20 

41 

1854 

9827 

2025 

9793 

2196 

9756 

2366 

9716 

2535 

9673 

19 

42 

1857 

9826 

2028 

9792 

2198 

9755 

2368 

9715 

2538 

9673 

18 

43 

1860 

9826 

2031 

9792 

2201 

9755 

2371 

9715 

2540 

9672 

17 

44 

1862 

9825 

2034 

9791 

2204 

9754 

2374 

9714 

2543 

9671 

16 

45 

1865 

9825 

2036 

9790 

2207 

9753 

2377 

9713 

2546 

9670 

15 

46 

1868 

9824 

2039 

9790 

2210 

9753 

2380 

9713 

2549 

9670 

14 

47 

1871 

9823 

2042 

9789 

2213 

9752 

2383 

9712 

2552 

9669 

13 

48 

1474 

9823 

2045 

9789 

2215 

9751 

2385 

9711 

2554 

9668 

12 

49 

1877 

9822 

2048 

9788 

2218 

9751 

2388 

9711 

2557 

9667 

11 

50 

1880 

9822 

2051 

9787 

2221 

9750 

2391 

9710 

2560 

9667 

10 

51 

1882 

9821 

2054 

9787 

2224 

9750 

2394 

9709 

2563 

9666 

9 

52 

1885 

9821 

2056 

9786 

2227 

9749 

2397 

9709 

2566 

9665 

8 

53 

1888 

9820 

2059 

9786 

2230 

9748 

2399 

9708 

2569 

9665 

7 

54 

1891 

9820 

2062 

9785 

2233 

9748 

2402 

9707 

2571 

9664 

6 

55 

1894 

9819 

2065 

9784 

2235 

9747 

2405 

9706 

2574 

9663 

,  5 

56 

1897 

9818 

2068 

9784 

2238 

9746 

240S 

9706 

2577 

9662 

4 

57 

1900 

9818 

2071 

9783 

2241 

9746 

2411 

9705 

2580 

9662 

3 

58 

1902 

9817 

2073 

9783 

2244 

9745 

2414 

9704 

2583 

9661 

2 

59 

1905 

9817 

2076 

9782 

2247 

9744 

2416 

9704 

2585 

9660 

1 

60 

1908 

9816 

2079 

9781 

2250 

9744 

2419 

9703 

2588 

9659 

0 

cos    sin 

79° 

cos    sin 

78° 

cos    sin 

77° 

cos 

sin 

cos    sin 

75° 

' 

76° 

r 

NATURAL 

SINES  AND 

COSINES. 

55 

f 

15° 

16° 

17° 

18° 

19° 

r. 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

o 

2588 

9659 

2756 

9613 

2924 

9563 

3090 

9511 

3256 

9455 

60 

1 

2591 

9659 

2759 

9612 

2926 

9562 

3093 

9510 

3258 

9454 

59 

2 

2594 

9658 

2762 

9611 

2929 

9561 

3096 

9509 

3261 

9453 

58 

3 

2597 

9657 

2765 

9610 

2932 

9560 

3098 

9508 

3264 

9452 

57 

4 

2599 

9656 

2768 

9609 

2935 

9560^ 

3101 

9507 

3267 

9451 

56 

5 

2602 

9655 

2770 

9609 

2938 

9559 

3104 

9506 

3269 

9450 

55 

6 

2605 

9655 

2773 

9608 

2940 

9558 

3107 

9505 

3272 

9449 

54 

7 

2608 

9654 

2776 

9607 

2943 

9557 

3110 

9504 

3275 

9449 

53 

8 

2611 

9653 

2779 

9606 

2946 

9556 

3112 

9503 

3278 

9448 

52 

9 

2613 

9652 

2782 

9605 

2949 

9555 

3115 

9502 

3280 

9447 

51 

10 

2616 

9652 

2784 

9605 

2952 

9555 

3118 

9502 

3283 

9446 

SO 

11 

2619 

9651 

2787 

9604 

2954 

9554 

3121 

9501 

3286 

9445 

49 

12 

2622 

9650 

2790 

9603 

2957 

9553 

3123 

9500 

3289 

9444 

48 

13 

2625 

9649 

2793 

9602 

2960 

9552 

3126 

9499 

3291 

9443 

47 

14 

2628 

9649 

2795 

9601 

2963 

9551 

3129 

9498 

3294 

9442 

46 

15 

2630 

9648 

2798 

9600 

2965 

9550 

3132 

9497 

3297 

9441 

45 

16 

2633 

9647 

2801 

9600 

2968 

9549 

3134 

9496 

3300 

9440 

44 

17 

2636 

9646 

2804 

9599 

2971 

9548 

3137 

9495 

3302 

9439 

43 

18 

2639 

9646 

2807 

9598 

2974 

9548 

3140 

9494 

3305 

9438 

42 

19 

2642 

9645 

2809 

9597 

2977 

9547 

3143 

9493 

3308 

9437 

41 

20 

2644 

9644 

2812 

9596 

2979 

9546 

3145 

9492 

3311 

9436 

40 

21 

2647 

9643 

2815 

9596 

2982 

9545 

3148 

9492 

3313 

9435 

39 

22 

2650 

9642 

2818 

9595 

2985 

9544 

3151 

9491 

3316 

9434 

38 

23 

2653 

9642 

2821 

9594 

2988 

9543 

3154 

9490 

3319 

9433 

37 

24 

2656 

9641 

2823 

9593 

2990 

9542 

3156 

9489 

3322 

9432 

36 

25 

2658 

9640 

2826 

9592 

2993 

9542 

3159 

9488 

3324 

9431 

35 

26 

2661 

9639 

2829 

9591 

2996 

9541 

3162 

9487 

3327 

9430 

34 

27 

2664 

9639 

2832 

9591 

2999 

9540 

3165 

9486 

3330 

9429 

33 

28 

2667 

9638 

2835 

9590 

3002 

9539 

3168 

9485 

3333 

9428 

32 

29 

2670 

9637 

2837 

9589 

3004 

9538 

3170 

9484 

3335 

9427 

31 

30 

2672 

9636 

2840 

9588 

3007 

9537 

3173 

9483 

3338 

9426 

30 

31 

2675 

9636 

2843 

9587 

3010 

9536 

3176 

9482 

3341 

9425 

29 

32 

2678 

9635 

2846 

9587 

3013 

9535 

3179 

9481 

3344 

9424 

28 

33 

2681 

9634 

2849 

9586 

3015 

9535 

3181 

9480 

3346 

9423 

27 

34 

2684 

9633 

2851 

9585 

3018 

9534 

3184 

9480 

3349 

9423 

26 

35 

26S6 

9632 

2854 

9584 

3021 

9533 

3187 

9479 

3352 

9422 

25 

36 

2689 

9632 

2857 

9583 

3024 

9532 

3190 

9478 

3355 

9421 

24 

37 

2692 

9631 

2860 

9582 

3026 

9531 

3192 

9477 

3357 

9420 

23 

38 

2695 

9630 

2862 

9582 

3029 

9530 

3195 

9476 

3360 

9419 

22 

39 

2698 

9629 

2865 

9581 

3032 

9529 

3198 

9475 

3363 

9418 

21 

40 

2700 

9628 

2868 

9580 

3035 

9528 

3201 

9474 

3365 

9417 

20 

41 

2703 

9628 

2871 

9579 

'  3038 

9527 

3203 

9473 

3368 

9416 

19 

42 

2706 

9627 

2874 

9578 

3040 

9527 

3206 

9472 

3371 

9415 

18 

43 

2709 

9626 

2876 

9577 

3043 

9526 

3209 

9471 

3374 

9414 

17 

44 

2712 

9625 

2879 

9577 

3046 

9525 

3212 

9470 

3376 

9413 

16 

45 

2714 

9625 

2882 

9576 

3049 

9524 

3214 

9469 

3379 

9412 

15 

46 

2717 

9624 

2885 

9575 

3051 

9523 

3217 

9468 

3382 

9411 

14 

47 

2720 

9623 

2888 

9574 

3054 

9522 

3220 

9467 

3385 

9410 

13 

48 

2723 

9622 

2890 

9573 

3057 

9521 

3223 

9466 

3387 

9409 

12 

49 

2726 

9621 

2893 

9572 

3060 

9520 

3225 

9466 

3390 

9408 

11 

50 

2728 

9621 

2896 

9572 

3062 

9520 

3228 

9465 

3393 

9407 

io 

51 

2731 

9620 

2899 

9571 

3065 

9519 

3231 

9464 

3396 

9406 

9 

52 

2734 

9619 

2901 

9570 

3068 

9518 

3234 

9463 

3398 

9405 

8 

53 

2737 

9618 

2904 

9569 

3071 

9517 

3236 

9462 

3401 

9404 

7 

54 

2740 

9617 

2907 

9568 

3074 

9516 

3239 

9461 

3404 

9403 

6 

55 

2742 

9617' 

2910 

9567 

3076 

9515 

3242 

9460 

3407 

9402 

5 

56. 

2745 

9616 

2913 

9566 

3079 

9514 

3245 

9459 

3409 

9401 

4 

57 

2748 

9615 

2915 

9566 

3082 

9513 

3247 

9458 

3412 

9400 

3 

58 

2751 

9614 

2918 

9565 

3085 

9512 

3250 

9457 

3415 

9399 

2 

59 

2754 

9613 

2921 

9564 

3087 

9511 

3253 

9456 

3417 

9398 

1 

60 

2756 

9613 

2924 

9563 

3090 

9511 

3256 

9455 

3420 

9397 

O 

cog   sin 

74° 

cos   sin 

73° 

cos   sin 

72° 

cos   sin 

71° 

cos    sin 

70° 

/ 

f 

56 


NATURAL   SINES  AND  COSINES. 


t 

20° 

sin   cog 

21° 

22° 

23° 

24° 

f 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

o 

3420 

9397 

3584 

9336 

3746 

9272 

3907 

9205 

4067 

9135 

60 

1 

3423 

9396 

3586 

9335 

3749 

9271 

3910 

9204 

4070 

9134 

59 

2 

3426 

9395 

3589 

9334 

3751 

9270 

3913 

9203 

4073 

9133 

58 

3 

3428 

9394 

3592 

9333 

3754 

9269 

3915 

9202 

4075 

9132 

57 

4 

3431 

9393 

3595 

9332 

3757 

9267 

3918 

9200 

4078 

9131 

56 

5 

3434 

9392 

3597 

9331 

3760 

9266 

3921 

9199 

4081 

9130 

55 

6 

3437 

9391 

3600 

9330 

3762 

9265 

3923 

9198 

4083 

9128 

54 

7 

3439 

9390 

3603 

9328 

3765 

9264 

3926 

9197 

4086 

9127 

53 

8 

3442 

9389 

3605 

9327 

3768 

9263 

3929 

9196 

4089 

9126 

52 

9 

3445 

9388 

3608 

9326 

3770 

9262 

3931 

9195 

4091 

9125 

51 

io 

3448 

9387 

3611 

9325 

3773 

9261 

3934 

9194 

4094 

9124 

SO 

11 

3450 

9386 

3614 

9324 

3776 

9260 

3937 

9192 

4097 

9122 

49 

12 

3453 

9385 

3616 

9323 

3778 

9259 

3939 

9191 

4099 

9121 

48 

13 

3456 

9384 

3619 

9322 

3781 

9258 

3942 

9190 

4102 

9120 

47 

14 

3458 

9383 

3622 

9321 

3784 

9257 

3945 

9189 

4105 

9119 

46 

15 

3461 

9382 

3624 

9320 

3786 

9255 

3947 

9188 

4107 

9118 

45 

16 

3464 

9381 

3627 

9319 

3789 

9254 

3950 

9187 

4110 

9116 

44 

17 

3467 

9380 

3630 

9318 

3792 

9253 

3953 

9186 

4112 

9115 

43 

18 

3469 

9379 

3633 

9317 

3795 

9252 

3955 

9184 

4115 

9114 

42 

19 

3472 

9378 

3635 

9316 

3797 

9251 

3958 

9183 

4118 

9113 

41 

20 

3475 

9377 

3638 

9315 

3800 

9250 

3961 

9182 

4120 

9112 

40 

21 

3478 

9376 

3641 

9314 

3803 

9249 

3963 

9181 

4123 

9110 

39 

22 

3480 

9375 

3643 

9313 

3805 

9248 

3966 

9180 

4126 

9109 

38 

23 

3483 

9374 

3646 

9312 

3808 

9247 

3969 

9179 

4128 

9108 

37 

24 

3486 

9373 

3649 

9311 

3811 

9245 

3971 

9178 

4131 

9107 

36 

25 

3488 

9372 

3651 

9309 

3813 

9244 

3974 

9176 

4134 

9106 

35 

26 

3491 

9371 

3654 

9308 

3816 

9243 

3977 

9175 

4136 

9104 

34 

27 

3494 

9370 

3657 

9307 

3819 

9242 

3979 

9174 

4139 

9103 

33 

28 

3497 

9369 

3660 

9306 

3821 

9241 

3982 

9173 

4142 

9102 

32 

29 

3499 

9368 

3662 

9305 

3824 

9240 

3985 

9172 

4144 

9101 

31 

30 

3502 

9367 

3665 

9304 

3827 

9239 

3987 

9171 

4147 

9100 

30 

31 

3505 

9366 

3668 

9303 

3830 

9238 

3990 

9169 

4150 

9098 

29 

32 

3508 

9365 

3670 

9302 

3832 

9237 

3993 

9168 

4152 

9097 

28 

33 

3510 

9364 

3673 

9301 

3835 

9235 

3995 

9167 

4155 

9096 

27 

34 

3513 

9363 

3676 

9300 

3838 

9234 

3998 

9166 

4158 

9095 

26 

35 

3516 

9362 

3679 

9299 

3840 

9233 

4001 

9165 

4160 

9094 

25 

36 

3518 

9361 

3681 

9298 

3843 

9232 

4003 

9164 

4163 

9092 

24 

37 

3521 

9360 

3684 

9297 

3846 

9231 

4006 

9162 

4165 

9091 

23 

38 

3524 

9359 

3687 

9296 

3848 

9230 

4009 

9161 

4168 

9090 

22 

39 

3527 

9358 

3689 

9295 

3851 

9229 

4011 

9160 

4171 

9088 

21 

40 

3529 

9356 

3692 

9293 

3854 

9228 

4014 

9159 

4173 

9088 

20 

41 

3532 

9355 

3695 

9292 

3856 

9227 

4017 

9158 

4176 

9086 

19 

42 

3535 

9354 

3697 

9291 

3859 

9225 

4019 

9157 

4179 

9085 

18 

43 

3537 

9353 

3700 

9290 

3862 

9224 

4022 

9155 

4181 

9084 

17 

44 

3540 

9352 

3703 

9289 

3864 

9223 

4025 

9154 

4184 

9083 

16 

45 

3543 

9351 

3706 

9288 

3867 

9222 

4027 

9153 

4187 

9081 

15 

46 

3546 

9350 

3708 

9287 

3870 

9221 

4030 

9152 

4189 

9080 

14 

47 

3548 

9349 

3711 

9286 

3872 

9220 

4033 

9151 

4192 

9079 

13 

48 

3551 

9348 

3714 

9285 

3875 

9219 

4035 

9150 

4195 

907S 

12 

49 

3554 

9347 

3716 

9284 

3878 

9218 

4038 

9148 

4197 

9077 

11 

50 

3557 

9346 

3719 

9283 

3881 

9216 

4041 

9147 

4200 

9075 

10 

51 

3559 

9345 

3722 

9282 

3883 

9215 

4043 

9146 

4202 

9074 

9 

52 

3562 

9344 

3724 

9281 

3886 

9214 

4046 

9145 

4205 

9073 

8 

53 

3565 

9343 

3727 

9279 

3889 

9213 

4049 

9144 

4208 

9072 

7 

54 

3567 

9342 

3730 

9278 

3891 

9212 

4051 

9143 

4210 

9070 

6 

55 

3570 

9341 

3733 

9277 

3894 

9211 

4054 

9141 

4213 

9069 

5 

56 

3573 

9340 

3735 

9276 

3897 

9210 

4057 

9140 

4216 

906S 

4 

57 

3576 

9339 

3738 

9275 

3899 

9208 

4059 

9139 

4218 

9067 

3 

58 

3578 

9338 

3741 

9274 

3902 

9207 

4062 

9138 

4221 

9066 

2 

59 

3581 

9337 

3743 

9273 

3905 

9206 

4065 

9137 

4224 

9064 

1 

60 

3584 

9336 

3746 

9272 

3907 

9205 

4067 

9135 

4226 

9063 

0 

cog   sin 

69° 

cos   sin 

68° 

cos   sin 

67° 

cos   sin 

66° 

cos   sin 

65° 

f 

t 

NATURAL  SINES   AND  COSINES.  57 


/ 

25° 

26° 

27° 

28° 

29° 

/ 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin   cos 

0 

4226 

9063 

4384 

8988 

4540 

8910 

4695 

8829 

4848  8746 

60 

1 

4229 

9062 

4386 

8987 

4542 

8909 

4697 

8828 

4851  8745 

59 

2 

4231 

9061 

4389 

8985 

4545 

8907 

4700 

8827 

4853  8743 

58 

3 

4234 

9059 

4392 

8984 

4548 

8906 

4702 

8825 

4856  8742 

57 

4 

4237 

9058 

4394 

8983 

4550 

8905 

4705 

8824 

4858  8741 

56 

5 

4239 

9057 

4397 

8982 

4553 

8903 

4708 

8823 

4861  8739 

55 

6 

4242 

9056 

4399 

8980 

4555 

8902 

4710 

8821 

4863  8738 

54 

7 

4245 

9054 

4402 

8979 

4558 

8901 

4713 

8820 

4866  8736 

53 

8 

4247 

9053 

4405 

8978 

4561 

8899 

4715 

8819 

4868  8735 

52 

9 

4250 

9052 

4407 

8976 

4563 

8898 

4718 

8817 

4871  8733 

51 

io 

4253 

9051 

4410 

8975 

4566 

8897 

4720 

8816 

4874  8732 

50 

11 

4255 

9050 

4412 

8974 

4568 

8895 

4723 

8814 

4876  8731 

49 

12 

4258 

9048 

4415 

8973 

4571 

8894 

4726 

8813 

4879  8729 

48 

13 

4260 

9047 

4418 

8971 

4574 

8893 

4728 

8812 

4881  8728 

47 

14 

4263 

9046 

4420 

8970 

4576 

8892 

4731 

8810 

4884  8726 

46 

15 

4266 

9045 

4423 

8969 

4579 

8890 

4733 

8809 

4886  8725 

45 

16 

4268 

9043 

4425 

8967 

4581 

8889 

4736 

8808 

4889  8724 

44 

17 

4271 

9042 

4428 

8966 

4584 

8888 

4738 

8806 

4891  8722 

43 

18 

4274 

9041 

4431 

8965 

4586 

8886 

4741 

8805 

4894  8721 

42 

19 

4276 

9040 

4433 

8964 

4589 

8885 

4743 

8803 

4896  8719 

41 

20 

4279 

9038 

4436 

8962 

4592 

8884 

4746 

8802 

4899  8718 

40 

21 

4281 

9037 

4439 

8961 

4594 

8882 

4749 

8801 

4901  8716 

39 

22 

4284 

9036 

4441 

8960 

4597 

8881 

4751 

8799 

4904  8715 

38 

23 

4287 

9035 

4444 

8958 

4599 

8879 

4754 

8798 

4907  8714 

37 

24 

4289 

9033 

4446 

8957 

4602 

8878 

4756 

8796 

4909  8712 

36 

25 

4292 

9032 

4449 

8956 

4605 

8877 

4759 

8795 

4912  8711 

35 

26 

4295 

9031 

4452 

8955 

4607 

8875 

4761 

8794 

4914  8709 

34 

27 

4297 

9030 

4454 

8953 

4610 

8874 

4764 

8792 

4917  8708 

33 

28 

4300 

9028 

4457 

8952 

4612 

8873 

4766 

8791 

4919  8706 

32 

29 

4302 

9027 

4459 

8951 

4615 

8871 

4769 

8790 

4922  8705 

31 

30 

4305 

9026 

4462 

8949 

4617 

8870 

4772 

8788 

4924  8704 

30 

31 

4308 

9025 

4465 

8948 

4620 

8869 

4774 

8787 

4927  8702 

29 

32 

4310 

9023 

4467 

8947 

4623 

8867 

4777 

8785 

4929  8701 

28 

33 

4313 

9022 

4470 

8945 

4625 

8866 

4779 

8784 

4932  8699 

27 

34 

4316 

9021 

4472 

8944 

4628 

8865 

4782 

8783 

4934  8698 

26 

35 

4318 

9020 

4475 

8943 

4630 

8863 

4784 

8781 

4937  8696 

25 

36 

4321 

9018 

4478 

8942 

4633 

8862 

4787 

8780 

4939  8695 

24 

37 

4323 

9017 

4480 

8940 

4636 

8861 

4789 

8778 

4942  8694 

23 

38 

4326 

9016 

4483 

8939 

4638 

8859 

4792 

8777 

4944  8692 

22 

39 

4329 

9015 

4485 

8938 

4641 

8858 

4795 

8776 

4947  8691 

21 

40 

4331 

9013 

4488 

8936 

4643 

8857 

4797 

8774 

4950  8689 

20 

41 

4334 

9012 

4491 

8935 

4646 

8855 

4800 

8773 

4952  8688 

19 

42 

4337 

9011 

4493 

8934 

4648 

8854 

4802 

8771 

4955  8686 

18 

43 

4339 

9010 

4496 

8932 

4651 

8853 

4805 

8770 

4957  8685 

17 

44 

4342 

9008 

4498 

8931 

4654 

8851 

4807 

8769 

4960  8683 

16 

45 

4344 

9007 

4501 

8930 

4656 

8850 

4810 

8767 

4962  8682 

15 

46 

4347 

9006 

4504 

8928 

4659 

8849 

4812 

8766 

4965  8681 

14 

47 

4350 

9004 

4506 

8927 

4661 

8847 

4815 

8764 

4967  8679 

13 

48 

4352 

9003 

4509 

8926 

4664 

8846 

4818 

8763 

4970  8678 

12 

49 

4355 

9002 

4511 

8925 

4666 

8844 

4820 

8762 

4972  8676 

11 

50 

4358 

9001 

4514 

8923 

4669 

8843 

4823 

8760 

4975  8675 

IO 

51 

4360 

8999 

4517 

8922 

4672 

8842 

4825 

8759 

4977  8673 

9 

52 

4363 

8998 

4519 

8921 

4674 

8840 

4828 

8757 

4980  8672 

8 

53 

4365 

8997 

4522 

8919 

4677 

8839 

4830 

8756 

4982  8670 

7 

54 

4368 

8996 

4524 

8918 

4679 

8838 

4833 

8755 

4985  8669 

6 

55 

4371 

8994 

4527 

8917 

4682 

8836 

4835 

8753 

4987  8668 

5 

56 

4373 

8993 

4530 

8915 

4684 

8835 

4838 

8752 

4990  8666 

4 

57 

4376 

8992 

4532 

8914 

4687 

8834 

4840 

8750 

4992  8665 

3 

58 

4378 

8990 

4535 

8913 

4690 

8832 

4843 

8749 

4995  8663 

2 

59 

4381 

8989 

4537 

8911 

4692 

8831 

4846 

8748 

4997  8662 

1 

OO 

4384 

8988 

4540 

8910 

4695 

8829 

4848 

8746 

5000  8660 

O 



cos    sin 

64° 

cos    sin 

63° 

cos    sin 

62° 

cos   sin 

61° 

cos   sin 

60° 

1' 

t 

58 

NATURAL 

SINES  AND 

COSINES. 

f 

30° 

31° 

32° 

33° 

34° 

f 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

O 

5000 

8660 

5150 

8572 

5299 

8480 

5446 

8387 

5592 

8290 

60 

1 

5003 

8659 

5153 

8570 

5302 

8479 

5449 

8385 

5594 

8289 

59 

2 

5005 

8657 

5155 

8569 

5304 

8477 

5451 

8384 

5597 

8287 

58 

3 

5008 

8656 

5158 

8567 

5307 

8476 

5454 

8382 

5599 

8285 

57 

4 

5010 

8654 

5160 

8566 

5309 

8474 

5456 

8380 

5602 

8284 

56 

5 

5013 

8653 

5163 

8564 

5312 

8473 

5459 

8379 

5604 

8282 

55 

6 

5015 

8652 

5165 

8563 

5314 

8471 

5461 

8377 

5606 

8281 

54 

7 

5018 

8650 

5168 

8561 

5316 

8470 

5463 

8376 

5609 

8279 

53 

8 

5020 

8649 

5170 

8560 

5319 

8468 

5466 

8374 

5611 

8277 

52 

9 

5023 

8647 

5173 

8558 

5321 

8467 

5468 

8372 

5614 

8276 

51 

10 

5025 

8646 

5175 

8557 

5324 

8465 

5471 

8371 

5616 

8274 

SO 

11 

5028 

8644 

5178 

8555 

5326 

8463 

5473 

8369 

5618 

8272 

49 

12 

5030 

8643 

5180 

8554 

5329 

8462 

5476 

8368 

5621 

8271 

48 

13 

5033 

8641 

5183 

8552 

5331 

8460 

5478 

8366 

5623 

8269 

47 

14 

5035 

8640 

5185 

8551 

5334 

8459 

5480 

8364 

5626 

8268 

46 

15 

5038 

8638 

5188 

8549 

5336 

8457 

5483 

8363 

5628 

8266 

45 

16 

5040 

8637 

5190 

8548 

5339 

8456 

5485 

8361 

5630 

8264 

44 

17 

5043 

8635 

5193 

8546 

5341 

8454 

5488 

8360 

5633 

8263 

43 

18 

5045 

8634 

5195 

8545 

5344 

8453 

5490 

8358 

5635 

8261 

42 

19 

5048 

8632 

5198 

8543 

5346 

8451 

5493 

8356 

5638 

8259 

41 

20 

5050 

8631 

5200 

8542 

5348 

8450 

5495 

8355 

5640 

8258 

40 

21 

5053 

8630 

5203 

8540 

5351 

8448 

5498 

8353 

5642 

8256 

39 

22 

5055 

8628 

5205 

8539 

5353 

8446 

5500 

8352 

5645 

8254 

38 

23 

5058 

8627 

5208 

8537 

5356 

8445 

5502 

8350 

5647 

8253 

37 

24 

5060 

8625 

5210 

8536 

5358 

8443 

5505 

8348 

5650 

8251 

36 

25 

5063 

8624 

5213 

8534 

5361 

8442 

5507 

8347 

5652 

8249 

35 

26 

5065 

8622 

5215 

8532 

5363 

8440 

5510 

8345 

5654 

8248 

34 

27 

5068 

8621 

5218 

8531 

5366 

8439 

5512 

8344 

5657 

8246 

33 

28 

5070 

8619 

5220 

8529 

5368 

8437 

5515 

8342 

5659 

8245 

32 

29 

5073 

8618 

5223 

8528 

5371 

8435 

5517 

8340 

5662 

8243 

31 

30 

5075 

8616 

5225 

8526 

5373 

8434 

5519 

8339 

5664 

8241 

30 

31 

5078 

8615 

5227 

8525 

5375 

8432 

5522 

8337 

5666 

8240 

29 

32 

5080 

8613 

5230 

8523 

5378 

8431 

5524 

8336 

5669 

8238 

28 

33 

5083 

8612 

5232 

8522 

5380 

8429 

5527 

8334 

5671 

8236 

27 

34 

5085 

8610 

5235 

8520 

5383 

8428 

5529 

8332 

5674 

8235 

26 

35' 

5088 

8609 

5237 

8519 

5385 

8426 

5531 

8331 

5676 

8233 

25 

36 

5090 

8607 

5240 

8517 

5388 

8425 

5534 

8329 

5678 

8231 

24 

37 

5093 

8606 

5242 

8516 

5390 

8423 

5536 

8328 

5681 

8230 

23 

38 

5095 

8604 

5245 

8514 

5393 

8421 

5539 

8326 

5683 

8228 

22 

39 

5098 

8603 

5247 

8513 

5395 

8420 

5541 

8324 

5686 

8226 

21 

40 

5100 

8601 

5250 

8511 

5398 

8418 

5544 

8323 

5688 

8225 

20 

41 

5103 

8600 

5252 

8510 

5400 

8417 

5546 

8321 

5690 

8223 

19 

42 

5105 

8599 

5255 

8508 

5402 

8415 

5548 

8320 

5693 

8221 

18 

43 

5108 

8597 

5257 

8507 

5405 

8414 

5551 

8318 

5695 

8220 

17 

44 

5110 

8596 

5260 

8505 

5407 

8412 

5553 

8316 

5698 

8218 

16 

45 

5113 

8594 

5262 

8504 

5410 

8410 

5556 

8315 

5700 

8216 

15 

46 

5115 

8593 

5265 

8502 

5412 

8409 

5558 

8313 

5702 

8215 

14 

47 

5118 

8591 

5267 

8500 

5415 

8407 

5561 

8311 

5705 

8213 

13 

48 

5120 

8590 

5270 

8499 

5417 

8406 

5563 

8310 

5707 

8211 

12 

49 

5123 

8588 

5272 

8497 

5420 

8404 

5565 

S30S 

5710 

8210 

11 

50 

5125 

8587 

5275 

8496 

5422 

8403 

5568 

8307 

5712 

8208 

io 

51 

5128 

8585 

5277 

8494 

5424 

8401 

5570 

8305 

5714 

8207 

9 

52 

5130 

8584 

5279 

8493 

5427 

8399 

5573 

8303 

5717 

8205 

8 

53 

5133 

8582 

5282 

8491 

5429 

8398 

5575 

8302 

5719 

8203 

7 

54 

5135 

8581 

5284 

8490 

5432 

8396 

5577 

8300 

5721 

8202 

6 

55 

5138 

8579 

5287 

8488 

5434 

8395 

5580 

8299 

5724 

8200 

5 

56 

5140 

8578 

5289 

8487 

5437 

8393 

5582 

8297 

5726 

8198 

4 

57 

5143 

8576 

5292 

8485 

5439 

8391 

5585 

8295 

5729 

8197 

3 

58 

5145 

8575 

5294 

8484 

5442 

8390 

5587 

8294 

5731 

8195 

2 

59 

5148 

8573 

5297 

8482 

5444 

8388 

5590 

8292 

5733 

8193 

1 

60 

5150 

8572 

5299 

8480 

5446 

8387 

5592 

8290 

5736 

8192 

O 

cos   sin 

59° 

cos 

sin 

cos   sin 

57° 

cos    sin 

56° 

cos   sin 

55° 

t 

58° 

/ 

NATURAL 

SINES 

AND 

COSINES. 

59 

t 

35° 

30° 

37° 

38° 

39° 

r 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

o 

5736 

8192 

5878 

8090 

6018 

7986 

6157 

7880 

6293 

7771 

60 

1 

5738 

8190 

5880 

8088 

6020 

7985 

6159 

7878 

6295 

7770 

59 

2 

5741 

8188 

5883 

8087 

6023 

7983 

6161 

7877 

629S 

7768 

58 

3 

5743 

8187 

5885 

8085 

6025 

7981 

6163 

7875 

6300 

7766 

57 

4 

5745 

8185 

5887 

8083 

6027 

7979 

6166 

7873 

6302 

7764 

56 

5 

5748 

8183 

5890 

8082 

6030 

7978 

6168 

7871 

6305 

7762 

55 

6 

5750 

8181 

5892 

8080 

6032 

7976 

6170 

7869 

6307 

7760 

54 

7 

5752 

8180 

5894 

8078 

6034 

7974 

6173 

7868 

6309 

7759 

53 

8 

5755 

8178 

5897 

8076 

6037 

7972 

6175 

7866 

6311 

7757 

52 

9 

5757 

8176 

5899 

8075 

6039 

7971 

6177 

7864 

6314 

7755 

51 

io 

5760 

8175 

5901 

8073 

6041 

7969 

6180 

7862 

6316 

7753 

SO 

11 

5762 

8173 

5904 

8071 

6044 

7967 

6182 

7860 

6318 

7751 

49 

12 

5764 

8171 

5906 

8070 

6046 

7965 

6184 

7859 

6320 

7749 

48 

13 

5767 

8170 

5908 

8068 

6048 

7964 

6186 

7857 

6323 

7748 

47 

14 

5769 

816S 

5911 

8066 

6051 

7962 

6189 

7855 

6325 

7746 

46 

15 

5771 

8166 

5913 

8064 

6053 

7960 

6191 

7853 

6327 

7744 

45 

16 

5774 

8165 

5915 

8063 

6055 

7958 

6193 

7851 

6329 

7742 

44 

17 

5776 

8163 

5918 

8061 

6058 

7956 

6196 

7850 

6332 

7740 

43 

18 

5779 

8161 

5920 

8059 

6060 

7955 

6198 

7848 

6334 

7738 

42 

19 

5781 

8160 

5922 

8058 

6062 

7953 

6200 

7846 

6336 

7737 

41 

20 

5783 

8158 

5925 

8056 

6065 

7951 

6202 

7844 

6338 

7735 

40 

21 

5786 

8156 

5927 

8054 

6067 

7950 

6205 

7842 

6341 

7733 

39 

22 

5788 

8155 

5930 

8052 

6069 

7948 

6207 

7841 

6343 

7731 

38 

23 

5790 

8153 

5932 

8051 

6071 

7946 

6209 

7839 

6345 

7729 

37 

24 

5793 

8151 

5934 

8049 

6074 

7944 

6211 

7837 

6347 

7727 

36 

25 

5795 

8150 

5937 

8047 

6076 

7942 

6214 

7835 

6350 

7725 

35 

26 

5798 

8148 

5939 

8045 

6078 

7941 

6216 

7833 

6352 

7724 

34 

27 

5S00 

8146 

5941 

8044 

6081 

7939 

6218 

7832 

6354 

7722 

33 

28 

5802 

8145 

5944 

8042 

6083 

7937 

6221 

7830 

6356 

7720 

32 

29 

5805 

8143 

5946 

8040 

6085 

7935 

6223 

7828 

6359 

7718 

31 

30 

5807 

8141 

5948 

8039 

6088 

7934 

6225 

7826 

6361 

7716 

30 

31 

5809 

8139 

•5951 

8037 

6090 

7932 

6227 

7824 

6363 

7714 

29 

32 

5812 

8138 

5953 

8035 

6092 

7930 

6230 

7822 

6365 

7713 

28 

33 

5814 

8136 

5955 

8033 

6095 

7928 

6232 

7821 

6368 

7711 

27 

34 

5S16 

8134 

5958 

8032 

6097 

7926 

6234 

7819 

6370 

7709 

26 

35 

5819 

8133 

5960 

8030 

6099 

7925 

6237 

7817 

6372 

7707 

25 

36 

5821 

8131 

5962 

8028 

6101 

7923 

6239 

7815 

6374 

7705 

24 

37 

5824 

8129 

5965 

8026 

6104 

7921 

6241 

7813 

6376 

7703 

23 

38 

5826 

8128 

5967 

8025 

6106 

7919 

6243 

7812 

6379 

7701 

22 

39 

5828 

8126 

5969 

8023 

6108 

7918 

6246 

7810 

6381 

7700 

21 

40 

5831 

8124 

5972 

8021 

6111 

7916 

6248 

7808 

6383 

7698 

20 

41 

5S33 

8123 

5974 

8020 

6113 

7914 

6250 

7806 

6385 

7696 

19 

42 

5835 

8121 

5976 

8018 

6115 

7912 

6252 

7804 

6388 

7694 

18 

43 

5838 

8119 

5979 

8016 

6118 

7910 

6255 

7802 

6390 

7692 

17 

44 

5840 

8117 

5981 

8014 

6120 

7909 

6257 

7801 

6392 

7690 

16 

45 

5842 

8116 

5983 

8013 

6122 

7907 

6259 

7799 

6394 

7688 

15 

46 

5845 

8114 

5986 

8011 

6124 

7905 

6262 

7797 

6397 

7687 

14 

47 

5847 

8112 

5988 

8009 

6127 

7903 

6264 

7795 

6399 

7685 

13 

48 

5850 

8111 

5990 

8007 

6129 

7902 

6266 

7793 

6401 

7683 

12 

49 

5852 

8109 

5993 

8006 

6131 

7900 

6268 

7792 

6403 

7681 

11 

SO 

5854 

8107 

5995 

8004 

6134 

7898 

6271 

7790 

6406 

7679 

IO 

51 

5857 

8106 

5997 

8002 

6136 

7896 

6273 

7788 

6408 

7677 

9 

52 

5859 

8104 

6000 

8000 

6138 

7894 

6275 

7786 

6410 

7675 

8 

53 

5861 

8102 

6002 

7999 

6141 

7893 

6277 

7784 

6412 

7674 

7 

54 

5864 

8100 

6004 

7997 

6143 

7891 

6280 

7782 

6414 

7672 

6 

55 

5866 

8099 

6007 

7995 

6145 

7889 

6282 

7781 

6417 

7670 

5 

56 

5868 

8097 

6009 

7993 

6147 

7887 

6284 

7779 

6419 

7668 

4 

57 

5871 

8095 

6011 

7992 

6150 

7885 

6286 

7777 

6421 

7666 

3 

58 

5873 

8094 

6014 

7990 

6152 

7884 

6289 

7775 

6423 

7664 

2 

59 

5875 

8092 

6016 

7988 

6154 

7882 

6291 

7773 

6426 

7662 

1 

60 

5878 

8090 

6018 

7986 

6157 

7880 

6293 

7771 

6428 

7660 

0 

cos    sin 

54° 

cos   sin 

53° 

cos    sin 

52° 

cos    sin 

51° 

cos    sin 

50° 

/ 

i 

60 

NATURAL 

SINES  AND 

COSINES. 

/ 

40° 

41° 

42° 

sin    cos 

43° 

44° 

/ 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

o 

6428 

7660 

6561 

7547 

6691 

7431 

6820 

7314 

6947 

7193 

60 

1 

6430 

7659 

6563 

7545 

6693 

7430 

6822 

7312 

6949 

7191 

59 

2 

6432 

7657 

6565 

7543 

6696 

7428 

6824 

7310 

6951 

7189 

58 

3 

6435 

7655 

6567 

7541 

6698 

7426 

6826 

7308 

6953 

7187 

57 

4 

6437 

7653 

6569 

7539 

6700 

7424 

6828 

7306 

6955 

7185 

56 

5 

6439 

7651 

6572 

7538 

6702 

7422 

6831 

7304 

6957 

7183 

55 

6 

6441 

7649 

6574 

7536 

6704 

7420 

6833 

7302 

6959 

7181 

54 

7 

6443 

7647 

6576 

7534 

6706 

7418 

6835 

7300 

6961 

7179 

53 

8 

6446 

7645 

6578 

7532 

6709 

7416 

6837 

7298 

6963 

7177 

52 

9 

6448 

7644 

6580 

7530 

6711 

7414 

6839 

7296 

6965 

7175 

51 

io 

6450 

7642 

6583 

7528 

6713 

7412 

6841 

7294 

6967 

7173 

50 

11 

6452 

7640 

6585 

7526 

6715 

7410 

6843 

7292 

6970 

7171 

49 

12 

6455 

7638 

6587 

7524 

6717 

7408 

6845 

7290 

6972 

7169 

48 

13 

6457 

7636 

6589 

7522 

6719 

7406 

6848 

7288 

6974 

7167 

47 

14 

6459 

7634 

6591 

7520 

6722 

7404 

6850 

7286 

6976 

7165 

46 

15 

6461 

7632 

6593 

7518 

6724 

7402 

6852 

7284 

6978 

7163 

45 

16 

6463 

7630 

6596 

7516 

6726 

7400 

6854 

7282 

6980 

7161 

44 

17 

6466 

7629 

6598 

7515 

6728 

7398 

6856 

7280 

6982 

7159 

43 

18 

6468 

7627 

6600 

7513 

6730 

7396 

6858 

7278 

6984 

7157 

42 

19 

6470 

7625 

6602 

7511 

6732 

7394 

6860 

7276 

6986 

7155 

41 

20 

6472 

7623 

6604 

7509 

6734 

7392 

6862 

7274 

6988 

7153 

40 

21 

6475 

7621 

6607 

7507 

6737 

7390 

6865 

7272 

6990 

7151 

39 

22 

6477 

7619 

6609 

7505 

6739 

7388 

6867 

7270 

6992 

7149 

38 

23 

6479 

7617 

66J1 

7503 

6741 

7387 

6869 

7268 

6995 

7147 

37 

24 

6481 

7615 

6613 

7501 

6743 

7385 

6871 

7266 

6997 

7145 

36 

25 

6483 

7613 

6615 

7499 

6745 

7383 

6873 

7264 

6999 

7143 

35 

26 

6486 

7612 

6617 

7497 

6747 

7381 

6875 

7262 

7001 

7141 

34 

27 

6488 

7610 

6620 

7495 

6749 

7379 

6877 

7260 

7003 

7139 

33 

28 

6490 

7608 

6622 

7493 

6752 

7377 

6879 

7258 

7005 

7137 

32 

29 

6492 

7606 

6624 

7491 

6754 

7375 

6881 

7256 

7007 

7135 

31 

30 

6494 

7604 

6626 

7490 

6756 

7373 

6884 

7254 

7009 

7133 

30 

31 

6497 

7602 

6628 

7488 

6758 

7371 

6886 

7252 

7011 

7130 

29 

32 

6499 

7600 

6631 

7486 

6760 

7369 

6888 

7250 

7013 

7128 

28 

33 

6501 

7598 

6633 

7484 

6762 

7367 

6890 

7248 

7015 

7126 

27 

34 

6503 

7596 

6635 

7482 

6764 

7365 

6892 

7246 

7017 

7124 

26 

35 

6506 

7595 

6637 

7480 

6767 

7363 

6894 

7244 

7019 

7122 

25 

36 

6508 

7593 

6639 

7478 

6769 

7361 

6896 

7242 

7022 

7120 

24 

37 

6510 

7591 

6641 

7476 

6771 

7359 

6898 

7240 

7024 

7118 

23 

38 

6512 

7589 

6644 

7474 

6773 

7357 

6900 

7238 

7026 

7116 

22 

39 

6514 

7587 

6646 

7472 

6775 

7355 

6903 

7236 

7028 

7114 

21 

40 

6517 

7585 

6648 

7470 

6777 

7353. 

6905 

7234 

7030 

7112 

20 

41 

6519 

7583 

6650 

7468 

6779 

7351 

6907 

7232 

7032 

7110 

19 

42 

6521 

7581 

6652 

7466 

6782 

7349 

6909 

7230 

7034 

7108 

18 

43 

6523 

7579 

6654 

7464 

6784 

7347 

6911 

7228 

7036 

7106 

17 

44 

6525 

7578 

6657 

7463 

6786 

7345 

6913 

7226 

7038 

7104 

16 

45 

6528 

7576 

6659 

7461 

6788 

7343 

6915 

7224 

7040 

7102 

15 

46 

6530 

7574 

6661 

7459 

6790 

7341 

6917 

7222 

7042 

7100 

14 

47 

6532 

7572 

6663 

7457 

6792 

7339 

6919 

7220 

7044 

7098 

13 

48 

6534 

7570 

6665 

7455 

6794 

7337 

6921 

7218 

7046 

7096 

12 

49 

6536 

7568 

6667 

7453 

6797 

7335 

6924 

7216 

7048 

7094 

11 

50 

6539 

7566 

6670 

7451 

6799 

7333 

6926 

7214 

7050 

7092 

10 

51 

6541 

7564 

6672 

7449 

6801 

7331 

6928 

7212 

7053 

7090 

9 

52 

6543 

7562 

6674 

7447 

6803 

7329 

6930 

7210 

7055 

7088 

8 

53 

6545 

7560 

6676 

7445 

6805 

7327 

6932 

7208 

7057 

7085 

7 

54 

6547 

7559 

6678 

7443 

6807 

7325 

6934 

7206 

7059 

7083 

6 

55 

6550 

7557 

6680 

7441 

6809 

7323 

6936 

7203 

7061 

7081 

5 

56 

6552 

7555 

6683 

7439 

6811 

7321 

6938 

7201 

7063 

7079 

4 

57 

6554 

7553 

6685 

7437 

6814 

7319 

6940 

7199 

7065 

7077 

3 

58 

6556 

7551 

6687 

7435 

6816 

7318 

6942 

7197 

7067 

7075 

2 

59 

6558 

7549 

6689 

7433 

6818 

7316 

6944 

7195 

7069 

7073 

1 

60 

6561 

7547 

6691 

7431 

6820 

7314 

6947 

7193 

7071 

7071 

O 

cos   sin 

49° 

cos    sin 

48° 

cos   sin 

47° 

cos 

sin 

cos   sin 

45° 

/ 

46° 

/ 

NATURAL  TANGENTS  AND 

COTANGENTS. 

61 

t 

0° 

1° 

2° 

3° 

4° 

t 

tan 

cot 

tan    cot 

tan 

cot 

tan    cot 

tan    cot 

o 

0000 

Infinite 

0175  57.2900 

0349 

28.6363 

0524  19.0811 

0699  14.3007 

60 

1 

0003 

3437.75 

0177  56.3506 

0352 

28.3994 

0527  18.9755 

0702  14.2411 

59 

2 

0006 

1718.87 

0180  55.4415 

0355 

28.1664 

0530  18.8711 

0705  14.1821 

58 

3 

0009 

1145.92 

0183  54.5613 

0358 

27.9372 

0533  18.7678 

0708  14.1235 

57 

4 

0012 

859.436 

0186  53.7086 

0361 

27.7117 

0536  18.6656 

0711  14.0655 

56 

5 

0015 

687.549 

0189  52.8821 

0364 

27.4899 

0539  18.5645 

0714  14.0079 

55 

6 

0017 

572.957 

0192  52.0807 

0367 

27.2715 

0542  18.4645 

0717  13.9507 

54 

7 

0020 

491.106 

0195  51.3032 

0370 

27.0566 

0544  18.3655 

0720  13.8940 

53 

8 

0023 

429.718 

0198  50.5485 

0373 

26.8450 

0547  18.2677 

0723  13.8378 

52 

9 

0026 

381.971 

0201  49.8157 

0375 

26.6367 

0550  18.1708 

0726  13.7821 

51 

io 

0029 

343.774 

0204  49.1039 

0378 

26.4316 

0553  18.0750 

0729  13.7267 

50 

11 

0032 

312.521 

0207  48.4121 

0381 

26.2296 

0556  17.9802 

0731  13.6719 

49 

12 

0035 

286.478 

0209  47.7395 

0384 

26.0307 

0559  17.8863 

0734  13.6174 

48 

13 

0038 

264.441 

0212  47.0853 

0387 

25.8348 

0562  17.7934 

0737  13.5634 

47 

14 

0041 

245.552 

0215  46.4489 

0390 

25.6418 

0565  17.7015 

0740  13.5098 

46 

15 

0044 

229.182 

0218  45.8294 

0393 

25.4517 

0568  17.6106 

0743  13.4566 

45 

16 

0047 

214.858 

0221  45.2261 

0396 

25.2644 

0571  17.5205 

0746  13.4039 

44 

17 

0049 

202.219 

0224  44.6386 

0399 

25.0798 

0574  17.4314 

0749  13.3515 

43 

18 

0052 

190.984 

0227  44.0661 

0402 

24.8978 

0577  17.3432 

0752  13.2996 

42 

19 

0055 

180.932 

0230  43.5081 

0405 

24.7185 

0580  17.2558 

0755  13.24S0 

41 

20 

0058 

171.885 

0233  42.9641 

0407 

24.5418 

0582  17.1693 

0758  13.1969 

40 

21 

0061 

163.700 

0236  42.4335 

0410 

24.3675 

0585  17.0837 

0761  13.1461 

39 

22 

0064 

156.259 

0239  41.9158 

0413 

24.1957 

0588  16.9990 

0764  13.0958 

38 

23 

0067 

149.465 

0241  41.4106 

0416 

24.0263 

0591  16.9150 

0767  13.0458 

37 

24 

0070 

143.237 

0244  40.9174 

0419 

23.8593 

0594  16.8319 

0769  12.9962 

36 

25 

0073 

137.507 

0247  40.4358 

0422 

23.6945 

0597  16.7496 

0772  12.9469 

35 

26 

0076 

132.219 

0250  39.9655 

0425 

23.5321 

0600  16.6681 

0775  12.8981 

34 

27 

0079 

127.321 

0253  39.5059 

0428 

23.3718 

0603  16.5874 

0778  12.8496 

33 

28 

0081 

122.774 

0256  39.0568 

0431 

23.2137 

0606  16.5075 

0781  12.8014 

32 

29 

0084 

118.540 

0259  38.6177 

0434 

23.0577 

0609  16.4283 

0784  12.7536 

31 

30 

0087 

114.589 

0262  38.1885 

0437 

22.9038 

0612  16.3499 

0787  12.7062 

30 

31 

0090 

110.892 

0265  37.7686 

0440 

22.7519 

0615  16.2722 

0790  12.6591 

29 

32 

0093 

107.426 

0268  37.3579 

0442 

22.6020 

0617  16.1952 

0793  12.6124 

28 

33 

0096 

104.171 

0271  36.9560 

0445 

22.4541 

0620  16.1190 

0796  12.5660 

27 

34 

0099 

101.107 

0274  36.5627 

0448 

22.3081 

0623  16.0435 

0799  12.5199 

26 

35 

0102 

98.2179 

0276  36.1776 

0451 

22.1640 

0626  15.9687 

0802  12.4742 

25 

36 

0105 

95.4895 

0279  35.8006 

0454 

22.0217 

0629  15.8945 

0805  12.4288 

24 

37 

0108 

92.9085 

0282  35.4313 

0457 

21.8813 

0632  15.8211 

0808  12.3838 

23 

38 

0111 

90.4633 

0285  35.0695 

0460 

21.7426 

0635  15.7483 

0810  12.3390 

22 

39 

0113 

88.1436 

0288  34.7151 

0463 

21.6056 

0638  15.6762 

0813  12.2946 

21 

40 

0116 

85.9398 

0291  34.3678 

0466 

21.4704 

0641  15.6048 

0816  12.2505 

20 

41 

0119 

83.8435 

0294  34.0273 

0469 

21.3369 

0644  15.5340 

0819  12.2067 

19 

42 

0122 

81.8470 

0297  33.6935 

0472 

21.2049 

0647  15.4638 

0822  12.1632 

18 

43 

0125 

79.9434 

0300  33.3662 

0475 

21.0747 

0650  15.3943 

0825  12.1201 

17 

44 

0128 

78.1263 

0303  33.0452 

0477 

20.9460 

0653  15.3254 

0828  12.0772 

16 

45 

0131 

76.3900 

0306  32.7303 

0480 

20.8188 

0655  15.2571 

0831  12.0346 

15 

46 

0134 

74.7292 

0308  32.4213 

0483 

20.6932 

0658  15.1893 

0834  11.9923 

14 

47 

0137 

73.1390 

0311  32.1181 

0486 

20.5691 

0661  15.1222 

0837  11.9504 

13 

48 

0140 

71.6151 

0314  31.8205 

0489 

20.4465 

0664  15.0557 

0840  11.9087 

12 

49 

0143 

70.1533 

0317  31.5284 

0492 

20.3253 

0667  14.9898 

0843  11.8673 

11 

50 

0146 

68.7501 

0320  31.2416 

0495 

20.2056 

0670  14.9244 

0846  11.8262 

10 

51 

0148 

67.4019 

0323  30.9599 

0498 

20.0872 

0673  14.8596 

0849  11.7853 

9 

52 

0151 

66.1055 

0326  30.6833 

0501 

19.9702 

0676  14.7954 

0851  11.7448 

8 

53 

0154 

64.8580 

0329  30.4116 

0504 

19.8546 

0679  14.7317 

0854  11.7045 

7 

54 

0157 

63.6567 

0332  30.1446 

0507 

19.7403 

0682  14.6685 

0857  11.6645 

6 

55 

0160 

62.4992 

0335  29.8823 

0509 

19.6273 

0685  14.6059 

0860  11.6248 

5 

56 

0163 

61.3829 

0338  29.6245 

0512 

19.5156 

0688  14.5438 

0863  11.5853 

4 

57 

0166 

60.3058 

0340  29.3711 

0515 

19.4051 

0690  14.4823 

0866  11.5461 

3 

58 

0169 

59.2659 

0343  29.1220 

0518 

19.2959 

0693  14.4212 

0869  11.5072 

2 

59 

0172 

58.2612 

0346  28.8771 

0521 

19.1879 

0696  14.3607 

0872  11.4685 

1 

60 

0175 

57.2900 

0349  28.6363 

0524 

19.0811 

0699  14.3007 

0875  11.4301 

0 

cot 

tan 

cot    tan 

cot 

tan 

cot    tan 

cot    tan 

/ 

89° 

88° 

87° 

86° 

85°     ' 

62 

NATURAL  TANGENTS  AND 

COTANGENTS 

/ 

5° 

6° 

7° 

8° 

9° 

/ 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

o 

0875 

11.4301 

1051 

9.5144 

1228 

8.1443 

1405 

7.1154 

1584 

6.3138 

60 

1 

0878 

11.3919 

1054 

9.4878 

1231 

8.1248 

1408 

7.1004 

1587 

6.3019 

59 

2 

0881 

11.3540 

1057 

9.4614 

1234 

8.1054 

1411 

7.0855 

1590 

6.2901 

58 

3 

0884 

11.3163 

1060 

9.4352 

1237 

8.0860 

1414 

7.0706 

1593 

6.2783 

57 

4 

0887 

11.2789 

1063 

9.4090 

1240 

8.0667 

1417 

7.0558 

1596 

6.2666 

56 

5 

0890 

11.2417 

1066 

9.3831 

1243 

8.0476 

1420 

7.0410 

1599 

6.2549 

55 

6 

0892 

11.2048 

1069 

9.3572 

1246 

8.0285 

1423 

7.0264 

1602 

6.2432 

54 

7 

0895 

11.1681 

1072 

9.3315 

1249 

8.0095 

1426 

7.0117 

1605 

6.2316 

53 

8 

0898 

11.1316 

1075 

9.3060 

1251 

7.9906 

1429 

6.9972 

1608 

6.2200 

52 

9 

0901 

11.0954 

1078 

92806 

1254 

7.9718 

1432 

6.9827 

1611 

6.2085 

51 

io 

0904 

11.0594 

1080 

9.2553 

1257 

7.9530 

1435 

6.9682 

1614 

6.1970 

50 

11 

0907 

11.0237 

1083 

9.2302 

1260 

7.9344 

1438 

6.9538 

1617 

6.1856 

49 

12 

0910 

10.9882 

1086 

9.2052 

1263 

7.9158 

1441 

6.9395 

1620 

6.1742 

48 

13 

0913 

10.9529 

1089 

9.1803 

1266 

7.8973 

1444 

6.9252 

1623 

6.1628 

47 

14 

0916 

10.9178 

1092 

9.1555 

1269 

7.8789 

1447 

6.9110 

1626 

6.1515 

46 

15 

0919 

10.8829 

1095 

9.1309 

1272 

7.8606 

1450 

6.8969 

1629 

6.1402 

45 

16 

0922 

]  0.8483 

1098 

9.1065 

1275 

7.8424 

1453 

6.8828 

1632 

6.1290 

44 

17 

0925 

10.8139 

1101 

9.0821 

1278 

7.8243 

1456 

6.8687 

1635 

6.1178 

43 

18 

0928 

10.7797 

1104 

9.0579 

1281 

7.8062 

1459 

6.8548 

1638 

6.1066 

42 

19 

0931 

10.7457 

1107 

9.0338 

1284 

7.7883 

1462 

6.8408 

1641 

6.0955 

41 

20 

0934 

10.7119 

1110 

9.0098 

1287 

7.7704 

1465 

6.8269 

1644 

6.0844 

40 

21 

0936 

10.6783 

1113 

8.9860 

1290 

7.7525 

1468 

6.8131 

1647 

6.0734 

39 

22 

0939 

10.6450 

1116 

8.9623 

1293 

7.7348 

1471 

6.7994 

1650 

6.0624 

38 

23 

0942 

10.6118 

1119 

8.9387 

1296 

7.7171 

1474 

6.7856 

1653 

6.0514 

37 

24 

0945 

10.5789 

1122 

8.9152 

1299 

7.6996 

1477 

6.7720 

1655 

6.0405 

36 

25 

0948 

10.5462 

1125 

8.8919 

1302 

7.6821 

1480 

6.7584 

1658 

6.0296 

35 

26 

0951 

10.5136 

1128 

8.8686 

1305 

7.6647 

1483 

6.7448 

1661 

6.0188 

34 

27 

0954 

10.4813 

1131 

8.8455 

1308 

7.6473 

1486 

6.7313 

1664 

6.0080 

33 

28 

0957 

10.4491 

1134 

8.8225 

1311 

7.6301 

1489 

6.7179 

1667 

5.9972 

32 

29 

0960 

10.4172 

1136 

8.7996 

1314 

7.6129 

1492 

6.7045 

1670 

5.9865 

31 

30 

0963 

10.3854 

1139 

8.7769 

1317 

7.5958 

1495 

6.6912 

1673 

5.9758 

30 

31 

0966 

10.3538 

1142 

8.7542 

1319 

7.5787 

1497 

6.6779 

1676 

5.9651 

29 

32 

0969 

10.3224 

1145 

8.7317 

1322 

7.5618 

1500 

6.6646 

1679 

5.9545 

28 

33 

0972 

10.2913 

1148 

8.7093 

1325 

7.5449 

1503 

6.6514 

1682 

5.9439 

27 

34 

0975 

10.2602 

1151 

8.6870 

1328 

7.5281 

1506 

6.6383 

1685 

5.9333 

26 

35 

0978 

10.2294 

1154 

8.6648 

1331 

7.5113 

1509 

6.6252 

1688 

5.9228 

25 

36 

0981 

10.1988 

1157 

8.6427 

1334 

7.4947 

1512 

6.6122 

1691 

5.9124 

24 

37 

0983 

10.1683 

1160 

8.6208 

1337 

7.4781 

1515 

6.5992 

1694 

5.9019 

23 

38 

0986 

10.1381 

1163 

8.5989 

1340 

7.4615 

1518 

6.5863 

1697 

5.8915 

22 

39 

0989 

10.1080 

1166 

8.5772 

1343 

7.4451 

1521 

6.5734 

1700 

5.8811 

21 

40 

0992 

10.0780 

1169 

8.5555 

1346 

7.4287 

1524 

6.5606 

1703 

5.8708 

20 

41 

0995 

10.0483 

1172 

8.5340 

1349 

7.4124 

1527 

6.5478 

1706 

5.8605 

19 

42 

0998 

10.0187 

1175 

8.5126 

1352 

7.3962 

1530 

6.5350 

1709 

5.8502 

18 

43 

1001 

9.9893 

1178 

8.4913 

1355 

7.3800 

1533 

6.5223 

1712 

5.8400 

17 

44 

1004 

9.9601 

1181 

8.4701 

1358 

7.3639 

1536 

6.5097 

1715 

5.8298 

16 

45 

1007 

9.9310 

1184 

8.4490 

1361 

7.3479 

1539 

6.4971 

1718 

5.8197 

15 

46 

1010 

9.9021 

1187 

8.4280 

1364 

7.3319 

1542 

6.4846 

1721 

5.8095 

14 

47 

1013 

9.8734 

1189 

8.4071 

1367 

7.3160 

1545 

6.4721 

1724 

5.7994 

13 

48 

1016 

9.8448 

1192 

8.3863 

1370 

7.3002 

1548 

6.4596 

1727 

5.7894 

12 

49 

1019 

9.8164 

1195 

8.3656 

1373 

7.2844 

1551 

6.4472 

1730 

5.7794 

11 

50 

1022 

9.7882 

1198 

8.3450 

1376 

7.2687 

1554 

6.4348 

1733 

5.7694 

IO 

51 

1025 

9.7601 

1201 

8.3245 

1379 

7.2531 

1557 

6.4225 

1736 

5.7594 

9 

52 

1028 

9.7322 

1204 

8.3041 

1382 

7.2375 

1560 

6.4103 

1739 

5.7495 

8 

53 

1030 

9.7044 

1207 

8.2838 

1385 

7.2220 

1563 

6.3980 

1742 

5.7396 

7 

54 

1033 

9.6768 

1210 

8.2636 

1388 

7.2066 

1566 

6.3859 

1745 

5.7297 

6 

55 

1036 

9.6499 

1213 

8.2434 

1391 

7.1912 

1569 

6.3737 

1748 

5.7199 

5 

56 

1039 

9.6220 

1216 

8.2234 

1394 

7.1759 

1572 

6.3617 

1751 

5.7101 

4 

57 

1042 

9.5949 

1219 

8.2035 

1397 

7.1607 

1575 

6.3496 

1754 

5.7004 

3 

58 

1045 

9.5679 

1222 

8.1837 

1399 

7.1455 

1578 

6.3376 

1757 

5.6906 

2 

59 

1048 

9.5411 

1225 

8.1640 

1402 

7.1304 

1581 

6.3257 

1760 

5.6809 

1 

60 

1051 

9.5144 

1228 

8.1443 

1405 

7.1154 

1584 

6.3138 

1763 

5.6713 

O 

r 

cot 

tan 

cot    tan 

83° 

cot    tan 

82° 

cot   tan 

81° 

cot    tan 

80° 

/ 

84° 

NATURAL  TANGENTS  AND 

COTANGENTS 

63 

/ 

io° 

11° 

12° 

13° 

14° 

f 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

o 

1763 

5.6713 

1944 

5.1446 

2126 

4.7046 

2309 

4.3315 

2493 

4.0108 

60 

1 

1766 

5.6617 

1947 

5.1366 

2129 

4.6979 

2312 

4.3257 

2496 

4.0058 

59 

2 

1769 

5.6521 

1950 

5.1286 

2132 

4.6912 

2315 

4.3200 

2499 

4.0009 

58 

3 

1772 

5.6425 

1953 

5.1207 

2135 

4.6845 

2318 

4.3143 

2503 

3.9959 

57 

4 

1775 

5.6330 

1956 

5.1128 

2138 

4.6779 

2321 

4.3086 

2506 

3.9910 

56 

5 

1778 

5.6234 

1959 

5.1049 

2141 

4.6712 

2324 

4.3029 

2509 

3.9861 

55 

6 

1781 

5.6140 

1962 

5.0970 

2144 

4.6646 

2327 

4.2972 

2512 

3.9812 

54 

7 

1784 

5.6045 

1965 

5.0892 

2147 

4.6580 

2330 

4.2916 

2515 

3.9763 

53 

8 

1787 

5.5951 

1968 

5.0814 

2150 

4.6514 

2333 

4.2859 

2518 

3.9714 

52 

9 

1790 

5.5857 

1971 

5.0736 

2153 

4.6448 

2336 

4.2803 

2521 

3.9665 

51 

10 

1793 

5.5764 

1974 

5.0658 

2156 

4.6382 

2339 

4.2747 

2524 

3.9617 

50 

11 

1796 

5.5671 

1977 

5.0581 

2159 

4.6317 

2342 

4.2691 

2527 

3.9568 

49 

12 

1799 

5.5578 

1980 

5.0504 

2162 

4.6252 

2345 

4.2635 

2530 

3.9520 

48 

13 

1802 

5.5485 

1983 

5.0427 

2165 

4.6187 

2349 

4.2580 

2533 

3.9471 

47 

14 

1805 

5.5393 

1986 

5.0350 

2168 

4.6122 

2352 

4.2524 

2537 

3.9423 

46 

15 

1808 

5.5301 

1989 

5.0273 

2171 

4.6057 

2355 

4.2468 

2540 

3.9375 

45 

16 

1811 

5.5209 

1992 

5.0197 

2174 

4.5993 

2358 

4.2413 

2543 

3.9327 

44 

17 

1814 

5.5118 

1995 

5.0121 

2177 

4.5928 

2361 

4.2358 

2546 

3.9279 

43 

18 

1817 

5.5026 

1998 

5.0045 

2180 

4.5864 

2364 

4.2303 

2549 

3.9232 

42 

19 

1820 

5.4936 

2001 

4.9969 

2183 

4.5800 

2367 

4.2248 

2552 

3.9184 

41 

20 

1823 

5.4845 

2004 

4.9894 

2186 

4.5736 

2370 

4.2193 

2555 

3.9136 

40 

21 

1826 

5.4755 

2007 

4.9819 

2189 

4.5673 

2373 

4.2139 

2558 

3.9089 

39 

22 

1829 

5.4665 

2010 

4.9744 

2193 

4.5609 

2376 

4.2084 

2561 

3.9042 

38 

23 

1832 

5.4575 

2013 

4.9669 

2196 

4.5546 

2379 

4.2030 

2564 

3.8995 

37 

24 

1835 

5.4486 

2016 

4.9594 

2199 

4.5483 

2382 

4.1976 

2568 

3.8947 

36 

25 

1838 

5.4397 

2019 

4.9520 

2202 

4.5420 

2385 

4.1922 

2571 

3.8900 

35 

26 

1841 

5.4308 

2022 

4.9446 

2205 

4.5357 

2388 

4.1868 

2574 

3.8854 

34 

27 

1844 

5.4219 

2025 

4.9372 

2208 

4.5294 

2392 

4.1814 

2577 

3.8807 

33 

28 

1847 

5.4131 

2028 

4.9298 

2211 

4.5232 

2395 

4.1760 

2580 

3.8760 

32 

29 

1850 

5.4043 

2031 

4.9225 

2214 

4.5169 

2398 

4.1706 

2583 

3.8714 

31 

30 

1853 

5.3955 

2035 

4.9152 

2217 

4.5107 

2401 

4.1653 

2586 

3.8667 

30 

31 

1856 

5.3868 

2038 

4.9078 

2220 

4.5045 

2404 

4.1600 

2589 

3.8621 

29 

32 

1859 

5.3781 

2041 

4.9006 

2223 

4.4983 

2407 

4.1547 

2592 

3.8575 

28 

33 

1862 

5.3694 

2044 

4.8933 

2226 

4.4922 

2410 

4.1493 

2595 

3.8528 

27 

34 

1865 

5.3607 

2047 

4.8860 

2229 

4.4860 

2413 

4.1441 

2599 

3.8482 

26 

35 

1868 

5.3521 

2050 

4.8788 

2232 

4.4799 

2416 

4.1388 

2602 

3.8436 

25 

36 

1871 

5.3435 

2053 

4.8716 

2235 

4.4737 

2419 

4.1335 

2605 

3.8391 

24 

37 

1874 

5.3349 

2056 

4.8644 

2238 

4.4676 

2422 

4.1282 

2608 

3.8345 

23 

38 

1877 

5.3263 

2059 

4.8573 

2241 

4.4615 

2425 

4.1230 

2611 

3.8299 

22 

39 

1880 

5.3178 

2062 

4.8501 

2244 

4.4555 

2428 

4.1178 

2614 

3.8254 

21 

40  1883 

5.3093 

2065 

4.8430 

2247 

4.4494 

2432 

4.1126 

2617 

3.8208 

20 

41 

1887 

5.3008 

2068 

4.8359 

2251 

4.4434 

2435 

4.1074 

2620 

3.8163 

19 

42 

1890 

5.2924 

2071 

4;8288 

2254 

4.4374 

2438 

4.1022 

2623 

3.8138 

18 

43 

1893 

5.2839 

2074 

4.8218 

2257 

4.4313 

2441 

4.0970 

2627 

3.8073 

17 

44 

1896 

5.2755 

2077 

4.8147 

2260 

4.4253 

2444 

4.0918 

2630 

3.8028 

16 

45 

1899 

5.2672 

2080 

4.8077 

2263 

4.4194 

2447 

4.0867 

2633 

3.7983 

15 

46 

1902 

5.2588 

2083 

4.8007 

2266 

4.4134 

2450 

4.0815 

2636 

3.7938 

14 

47 

1905 

5.2505 

2086 

4.7937 

2269 

4.4075 

2453 

4.0764 

2639 

3.7893 

13 

48 

1908 

5.2422 

2089 

4.7867 

2272 

4.4015 

2456 

4.0713 

2642 

3.7848 

12 

49 

1911 

5.2339 

2092 

4.7798 

2275 

4.3956 

2459 

4.0662 

2645 

3.7804 

11 

SO 

1914 

5.2257 

2095 

4.7729 

2278 

4.3897 

2462 

4.0611 

2648 

3.7760 

10 

51 

1917 

5.2174 

2098 

4.7659 

2281 

4.3838 

2465 

4.0560 

2651 

3.7715 

9 

52 

1920 

5.2092 

2101 

4.7591 

2284 

4.3779 

2469 

4.0509 

2655 

3.7671 

8 

53 

1923 

5.2011 

2104 

4.7522 

2287 

4.3721 

2472 

4.0459 

2658 

3.7627 

7 

54 

1926 

5.1929 

2107 

4.7453 

2290 

4.3662 

2475 

4.0408 

2661 

3.7583 

6 

55 

1929 

5.1848 

2110 

4.7385 

2293 

4.3604 

2478 

4.0358 

2664 

3.7539 

5 

56 

1932 

5.1767 

2113 

4.7317 

2296 

4.3546 

2481 

4.0308 

2667 

3.7495 

4 

57 

1935 

5.1686 

2116 

4.7249 

2299 

4.3488 

2484 

4.0257 

2670 

3.7451 

3 

58 

1938 

5.1606 

2119 

4.7181 

2303 

4.3430 

2487 

4.0207 

2673 

3.7408 

2 

59 

1941 

5.1526 

2123 

4.7114 

2306 

4.3372 

2490 

4.0158 

2676 

3.7364 

1 

60 

1944 

5.1446 

2126 

4.7046 

2309 

4.3315 

2493 

4.0108 

2679 

3.7321 

O 

t 

cot   tan 

79° 

cot    tan 

78° 

cot 

tan 

cot   tan 

76° 

cot    tan 

75° 

f 

77° 

64 

NATURAL  TANGENTS  AND 

COTANGENTS. 

r 

15° 

16° 

17° 

18° 

19° 

t 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

O 

2679 

3.7321 

2867 

3.4874 

3057 

3.2709 

3249 

3.0777 

3443 

2.9042 

60 

1 

2683 

3.7277 

2871 

3.4836 

3060 

3.2675 

3252 

3.0746 

3447 

2.9015 

59 

2 

2686 

3.7234 

2874 

3.4798 

3064 

3.2641 

3256 

3.0716 

3450 

2.8987 

58 

3 

2689 

3.7191 

2877 

3.4760 

3067 

3.2607 

3259 

3.0686 

3453 

2.8960 

57 

4 

2692 

3.7148 

2880 

3.4722 

3070 

3.2573 

3262 

3.0655 

3456 

2.8933 

56 

5 

2695 

3.7105 

2883 

3.4684 

3073 

3.2539 

3265 

3.0625 

3460 

2.8905 

55 

6 

2698 

3.7062 

2886 

3.4646 

3076 

3.2506 

3269 

3.0595 

3463 

2.8878 

54 

7 

2701 

3.7019 

2890 

3.4608 

3080 

3.2472 

3272 

3.0565 

3466 

2.8851 

53 

8 

2704 

3.6976 

2893 

3.4570 

3083 

3.2438 

3275 

3.0535 

3469 

2.8824 

52 

9 

2708 

3.6933 

2896 

3.4533 

3086 

3.2405 

3278 

3.0505 

3473 

2.8797 

51 

io 

2711 

3.6891 

2899 

3.4495 

3089 

3.2371 

3281 

3.0475 

3476 

2.8770 

50 

11 

2714 

3.6848 

2902 

3.4458 

3092 

3.2338 

3285 

3.0445 

3479 

2.8743 

49 

12 

2717 

3.6806 

2905 

3.4420 

3096 

3.2305 

3288 

3.0415 

3482 

2.8716 

48 

13 

2720 

3.6764 

2908 

3.4383 

3099 

3.2272 

3291 

3.0385 

3486 

2.8689 

47 

14 

2723 

3.6722 

2912 

3.4346 

3102 

3.2238 

3294 

3.0356 

3489 

2.8662 

46 

15 

2726 

3.6680 

2915 

3.4308 

3105 

3.2205 

3298 

3.0326 

3492 

2.8636 

45 

16 

2729 

3.6638 

2918 

3.4271 

3108 

3.2172 

3301 

3.0296 

3495 

2.8609 

44 

17 

2733 

3.6596 

2921 

3.4234 

3111 

3.2139 

3304 

3.0267 

3499 

2.8582 

43 

18 

2736 

3.6554 

2924 

3.4197 

3115 

3.2106 

3307 

3.0237 

3502 

2.8556 

42 

19 

2739 

3.6512 

2927 

3.4160 

3118 

3.2073 

3310 

3.0208 

3505 

2.8529 

41 

20 

2742 

3.6470 

2931 

3.4124 

3121 

3.2041 

3314 

3.0178 

3508 

2.8502 

40 

21 

2745 

3.6429 

2934 

3.4087 

3124 

3.2008 

3317 

3.0149 

3512 

2.8476 

39 

22 

2748 

3.6387 

2937 

3.4050 

3127 

3.1975 

3320 

3.0120 

3515 

2.8449 

38 

23 

2751 

3.6346 

2940 

3.4014 

3131 

3.1943 

3323 

3.0090 

3518 

2.8423 

37 

24 

2754 

3.6305 

2943 

3.3977 

3134 

3.1910 

3327 

3.0061 

3522 

2.8397 

36 

25 

2758 

3.6264 

2946 

3.3941 

3137 

3.1878 

3330 

3.0032 

3525 

2.8370 

35 

26 

2761 

3.6222 

2949 

3.3904 

3140 

3.1845 

3333 

3.0003 

3528 

2.8344 

34 

27 

2764 

3.6181 

2953 

3.3868 

3143 

3.1813 

3336 

2.9974 

3531 

2.8318 

33 

28 

2767 

3.6140 

2956 

3.3832 

3147 

3.1780 

3339 

2.9945 

3535 

2.8291 

32 

29 

2770 

3.6100 

2959 

3.3796 

3150 

3.1748 

3343 

2.9916 

3538 

2.8265 

31 

30 

2773 

3.6059 

2962 

3.3759 

3153 

3.1716 

3346 

2.9887 

3541 

2.8239 

30 

31 

2776 

3.6018 

2965 

3.3723 

3156 

3.1684 

3349 

2.9858 

3544 

2.8213 

29 

32 

2780 

3.5978 

2968 

3.3687 

3159 

3.1652 

3352 

2.9829 

3548 

2.8187 

28 

33 

2783 

3.5937 

2972 

3.3652 

3163 

3.1620 

3356 

2.9800 

3551 

2.8161 

27 

34 

2786 

3.5897 

2975 

3.3616 

3166 

3.1588 

3359 

2.9772 

3554 

2.8135 

26 

35 

2789 

3.5856 

2978 

3.3580 

3169 

3.1556 

3362 

2.9743 

3558 

2.8109 

25 

36 

2792 

3.5816 

2981 

3.3544 

3172 

3.1524 

3365 

2.9714 

3561 

2.8083 

24 

37 

2795 

3.5776 

2984 

3.3509 

3175 

3.1492 

3369 

2.9686 

3564 

2.8057 

23 

38 

2798 

3.5736 

2987 

3.3473 

3179 

3.1460 

3372 

2.9657 

3567 

2.8032 

22 

39 

2801 

3.5696 

2991 

3.3438 

3182 

3.1429 

3375 

2.9629 

3571 

2.8006 

21 

40 

2805 

3.5656 

2994 

3.3402 

3185 

3.1397 

3378 

2.9600 

3574 

2.7980 

20 

41 

2808 

3.5616 

2997 

3.3367 

3188 

3.1366 

3382 

2.9572 

3577 

2.7955 

19 

42 

2811 

3.5576 

3000 

3.3332 

3191 

3.1334 

3385 

2.9544 

3581 

2.7929 

18 

43 

2814 

3.5536 

3003 

3.3297 

3195 

3.1303 

3388 

2.9515 

3584 

2.7903 

17 

44 

2817 

3.5497 

3006 

3.3261 

3198 

3.1271 

3391 

2.9487 

3587 

2.7878 

16 

45 

2820 

3.5457 

3010 

3.3226 

3201 

3.1240 

3395 

2.9459 

3590 

2.7852 

15 

46 

2823 

3.5418 

3013 

3.3191 

3204 

3.1209 

3398 

2.9431 

3594 

2.7827 

14 

47 

2827 

3.5379 

3016 

3.3156 

3207 

3.1178 

3401 

2.9403 

3597 

2.7801 

13 

48 

2830 

3.5339 

3019 

3.3122 

3211 

3.1146 

3404 

2.9375 

3600 

2.7776 

12 

49 

2833 

3.5300 

3022 

3.3087 

3214 

3.1115 

3408 

2.9347 

3604 

2.7751 

11 

50 

2836 

3.5261 

3026 

3.3052 

3217 

3.1084 

3411 

2.9319 

3607 

2.7725 

IO 

51 

2839 

3.5222 

3029 

3.3017 

3220 

3.1053 

3414 

2.9291 

3610 

2.7700 

9 

52 

2842 

3.5183 

3032 

3.2983 

3223 

3.1022 

3417 

2.9263 

3613 

2.7675 

8 

53 

2845 

3.5144 

3035 

3.2948 

3227 

3.0991 

3421 

2.9235 

3617 

2.7650 

7 

54 

2849 

3.5105 

3038 

3.2914 

3230 

3.0961 

3424 

2.9208 

3620 

2.7625 

6 

55 

2852 

3.5067 

3041 

3.2880 

3233 

3.0930 

3427 

2.9180 

3623 

2.7500 

5 

56 

2855 

3.5028 

3045 

3.2845 

3236 

3.0899 

3430 

2.9152 

3627 

2.7575 

4 

57 

2858 

3.4989 

3048 

3.2811 

3240 

3.0868 

3434 

2.9125 

3630 

2.7550 

3 

58 

2861 

3.4951 

3051 

3.2777 

3243 

3.0838 

3437 

2.9097 

3633 

2.7525 

2 

59 

2864 

3.4912 

3054 

3.2743 

3246 

3.0807 

3440 

2.9070 

3636 

2.7500 

1 

60 

2867 

3.4874 

3057 

3.2709 

3249 

3.0777 

3443 

2.9042 

3640 

2.7475 

O 

/ 

cot    tan 

74° 

cot    tan 

73° 

cot 

tan 

cot    tan 

71° 

cot   tan 

70° 

t 

72° 

NATURAL  TANGENTS  AND 

COTANGENTS 

65 

t 

20° 

21° 

22° 

23° 

24° 

t 

tan   cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

o 

3640  2.7475 

3839 

2.6051 

4040 

2.4751 

4245 

2.3559 

4452 

2.2460 

60 

1 

3643  2.7450 

3842 

2.6028 

4044 

2.4730 

4248 

2.3539 

4456 

2.2443 

59 

2 

3646  2.7425 

3845 

2.6006 

4047 

2.4709 

4252 

2.3520 

4459 

2.2425 

58 

3 

3650  2.7400 

3849 

2.5983 

4050 

2.4689 

4255 

2.3501 

4463 

2,2408 

57 

4 

3653  2.7376 

3852 

2.5961 

4054 

2.4668 

4258 

2.3483 

4466 

2.2390 

56 

5 

3656  2.7351 

3855 

2.5938 

4057 

2.4648 

4262 

2.3464 

4470 

2.2373 

55 

6 

3659  2.7326 

3859 

2.5916 

4061 

2.4627 

4265 

2.3445 

4473 

2.2355 

54 

7 

3663  2.7302 

3862 

2.5893 

4064 

2.4606 

4269 

2.3426 

4477 

2.2338 

53 

8 

3666  2.7277 

3865 

2.5871 

4067 

2.4586 

4272 

2.3407 

4480 

2.2320 

52 

9 

3669  2.7253 

3869 

2.5848 

4071 

2.4566 

4276 

2.3388 

4484 

2.2303 

51 

io 

3673  2.7228 

3872 

2.5826 

4074 

2.4545 

4279 

2.3369 

4487 

2.2286 

50 

11 

3676  2.7204 

3875 

2.5804 

4078 

2.4525 

4283 

2.3351 

4491 

2.2268 

49 

12 

3679  2.7179 

3879 

2.5782 

4081 

2.4504 

4286 

2.3332 

4494 

2.2251 

48 

13 

3683  2.7155 

3882 

2.5759 

4084 

2.4484 

4289 

2.3313 

4498 

2.2234 

47 

14 

3686  2.7130 

3885 

2.5737 

4088 

2.4464 

4293 

2.3294 

4501 

2.2216 

46 

15 

3689  2.7106 

3889 

2.5715 

4091 

2.4443 

4296 

2.3276 

4505 

2.2199 

45 

16 

3693  2.7082 

3892 

2.5693 

4095 

2.4423 

4300 

2.3257 

4508 

2.2182 

44 

17 

3696  2.7058 

3895 

2.5671 

4098 

2.4403 

4303 

2.3238 

4512 

2.2165 

43 

18 

3699  2.7034 

3899 

2.5649 

4101 

2.4383 

4307 

2.3220 

4515 

2.2148 

42 

19 

3702  2.7009 

3902 

2.5627 

4105 

2.4362 

4310 

2.3201 

4519 

2.2130 

41 

20 

3706  2.6985 

3906 

2.5605 

4108 

2.4342 

4314 

2.3183 

4522 

2.2113 

40 

21 

3709  2.6961 

3909 

2.5583 

4111 

2.4322 

4317 

2.3164 

4526 

2.2096 

39 

22 

3712  2.6937 

3912 

2.5561 

4115 

2.4302 

4320 

2.3146 

4529 

2.2079 

38 

23 

3716  2.6913 

3916 

2.5539 

4118 

2.4282 

4324 

2.3127 

4533 

2.2062 

37 

24 

3719  2.6889 

3919 

2.5517 

4122 

2.4262 

4327 

2.3109 

4536 

2.2045 

36 

25 

3722  2.6865 

3922 

2.5495 

4125 

2.4242 

4331 

2.3090 

4540 

2.2028 

35 

26 

3726  2.6841 

3926 

2.5473 

4129 

2.4222 

4334 

2.3072 

4543 

2.2011 

34 

27 

3729  2.6818 

3929 

2.5452 

4132 

2.4202 

4338 

2.3053 

4547 

2.1994 

33 

28 

3732  2.6794 

3932 

2.5430 

4135 

2.4182 

4341 

2.3035 

4550 

2.1977 

32 

29 

3736  2.6770 

3936 

2.5408 

4139 

2.4162 

4345 

2.3017 

4554 

2.1960 

31 

30 

3739  2.6746 

3939 

2.53S6 

4142 

2.4142 

4348 

2.2998 

4557 

2.1943 

30 

31 

3742  2.6723 

3942 

2.5365 

4146 

2.4122 

4352 

2.2980 

4561 

2.1926 

29 

32 

3745  2.6699 

3946 

2.5343 

4149 

2.4102 

4355 

2.2962 

4564 

2.1909 

28 

33 

3749  2.6675 

3949 

2.5322 

4152 

2.4083 

4359 

2.2944 

4568 

2.1892 

27 

34 

3752  2.6652 

3953 

2.5300 

4156 

2.4063 

4362 

2.2925 

4571 

2.1876 

26 

35 

3755  2.6628 

3956 

2.5279 

4159 

2.4043 

4365 

2.2907 

4575 

2.1859 

25 

36 

3759  2.6605 

3959 

2.5257 

4163 

2.4023 

4369 

2.2889 

4578 

2.1842 

24 

37 

3762  2.6581 

3963 

2.5236 

4166 

2.4004 

4372 

2.2871 

4582 

2.1825 

23 

38 

3765  2.6558 

3966 

2.5214 

4169 

2.3984 

4376 

2.2853 

4585 

2.1808 

22 

39 

3769  2.6534 

3969 

2.5193 

4173 

2.3964 

4379 

2.2835 

4589 

2.1792 

21 

40 

3772  2.6511 

3973 

2.5172 

4176 

2.3945 

4383 

2.2817 

4592 

2.1775 

20 

41 

3775  2.6488 

3976 

2.5150 

4180 

2.3925 

4386 

2.2799 

4596 

2.1758 

19 

42 

3779  2.6464 

3979 

2.5129 

4183 

2.3906 

4390 

2.2781 

4599 

2.1742 

18 

43 

3782  2.6441 

3983 

2.5108 

4187 

2.3886 

4393 

2.2763 

4603 

2.1725 

17 

44 

3785  2.6418 

3986 

2.5086 

4190 

2.3867 

4397 

2.2745 

4607 

2.1708 

16 

45 

3789  2.6395 

3990 

2.5065 

4193 

2.3847 

4400 

2.2727 

4610 

2.1692 

15 

46 

3792  2.6371 

3993 

2.5044 

4197 

2.3828 

4404 

2.2709 

4614 

2.1675 

14 

47 

3795  2.6348 

3996 

2.5023 

4200 

2.3808 

4407 

2.2691 

4617 

2.1659 

13 

48 

3799  2.6325 

4000 

2.5002 

4204 

2.3789 

4411 

2.2673 

4621 

2.1642 

12 

49 

3802  2.6302 

4003 

2.4981 

4207 

2.3770 

4414 

2.2655 

4624 

2.1625 

11 

50 

3805  2.6279 

4006 

2.4960 

4210 

2.3750 

4417 

2.2637 

4628 

2.1609 

IO 

51 

3809  2.6256 

4010 

2.4939 

4214 

2.3731 

4421 

2.2620 

4631 

2.1592 

9 

52 

3812  2.6233 

4013 

2.4918 

4217 

2.3712 

4424 

2.2602 

4635 

2.1576 

8 

53 

3815  2.6210 

4017 

2.4897 

4221 

2.3693 

4428 

2.2584 

4638 

2.1560 

7 

54 

3819  2.6187 

4020 

2.4876 

4224 

2.3673 

4431 

2.2566 

4642 

2.1543 

6 

55 

3822  2.6165 

4023 

2.4855 

4228 

2.3654 

4435 

2.2549 

4645 

2.1527 

5 

56 

3825  2.6142 

4027 

2.4834 

4231 

2.3635 

4438 

2.2531 

4649 

2.1510 

4 

57 

3829  2.6119 

4030 

2.4813 

4234 

2.3616 

4442 

2.2513 

4652 

2.1494 

3 

58 

3832  2.6096 

4033 

2.4792 

4238 

2.3597 

4445 

2.2496 

4656 

2.1478 

2 

59 

3835  2.6074 

4037 

2.4772 

4241 

2.3578 

4449 

2.2478 

4660 

2.1461 

1 

60 

3839  2.6051 

4040 

2.4751 

4245 

2.3559 

4452 

2.2460 

4663 

2.1445 

0 

t 

cot    tan 

69° 

cot   tan 

68° 

cot    tan 

67° 

cot   tan 

66° 

cot   tan 

65° 

r 

66 

NATURAL  TANGENTS  AND 

COTANGENTS. 

/ 

25° 

26° 

27° 

28° 

29° 

t 

tan 

cot 

„  tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

O 

4663 

2.1445 

4877 

2.0503 

5095 

1.9626 

5317 

1.8807 

5543 

1.8040 

60 

1 

4667 

2.1429 

4881 

2.0488 

5099 

1.9612 

5321 

1.8794 

5547 

1.8028 

59 

2 

4670 

2.1413 

4885 

2.0473 

5103 

1.9598 

5325 

1.8781 

5551 

1.8016 

58 

3 

4674 

2.1396 

4888 

2.0458 

5106 

1.9584 

5328 

1.8768 

5555 

1.8003 

57 

4 

4677 

2.1380 

4892 

2.0443 

5110 

1.9570 

5332 

1.8755 

5558 

1.7991 

56 

5 

4681 

2.1364 

4895 

2.0428 

5114 

1.9556 

5336 

1.8741 

5562 

1.7979 

55 

6 

4684 

2.1348 

4899 

2.0413 

5117 

1.9542 

5340 

1.8728 

5566 

1.7966 

54 

7 

4688 

2.1332 

4903 

2.0398 

5121 

1.9528 

5343 

1.8715 

5570 

1.7954 

53 

8 

4691 

2.1315 

4906 

2.0383 

5125 

1.9514 

5347 

1.8702 

5574 

1.7942 

52 

9 

4695 

2.1299 

4910 

2.0368 

5128 

1.9500 

5351 

1.8689 

5577 

1.7930 

51 

io 

4699 

2.1283 

4913 

2.0353 

5132 

1.9486 

5354 

1.8676 

5581 

1.7917 

SO 

11 

4702 

2.1267 

4917 

2.0338 

5136 

1.9472 

5358 

1.8663 

5585 

1.7905 

49 

12 

4706 

2.1251 

4921 

2.0323 

5139 

1.9458 

5362 

1.8650 

5589 

1.7893 

48 

13 

4709 

2.1235 

4924 

2.0308 

5143 

1.9444 

5366 

1.8637 

5593 

1.7881 

47 

14 

4713 

2.1219 

4928 

2.0293 

5147 

1.9430 

5369 

1.8624 

5596 

1.7868 

46 

15 

4716 

2.1203 

4931 

2.0278 

5150 

1.9416 

5373 

1.8611 

5600 

1.7856 

45 

16 

4720 

2.1187 

4935 

2.0263 

5154 

1.9402 

5377 

1.8598 

5604 

1.7844 

44 

17 

4723 

2.1171 

4939 

2.0248 

5158 

1.9388 

5381 

1.8585 

5608 

1.7832 

43 

18 

4727 

2.1155 

4942 

2.0233 

5161 

1.9375 

5384 

1.8572 

5612 

1.7820 

42 

19 

4731 

2.1139 

4946 

2.0219 

5165 

1.9361 

5388 

1.8559 

5616 

1.7808 

41 

20 

4734 

2.1123 

4950 

2.0204 

5169 

1.9347 

5392 

1.8546 

5619 

1.7796 

40 

21 

4738 

2.1107 

4953 

2.0189 

5172 

1.9333 

5396 

1.8533 

5623 

1.7783 

39 

22 

4741 

2.1092 

4957 

2.0174 

5176 

1.9319 

5399 

1.8520 

5627 

1.7771 

38 

23 

4745 

2.1076 

4960 

2.0160 

5180 

1.9306 

5403 

1.8507 

5631 

1.7759 

37 

24 

4748 

2.1060 

4964 

2.0145 

5184 

1.9292 

5407 

1.8495 

5635 

1.7747 

36 

25 

4752 

2.1044 

4968 

2.0130 

5187 

1.9278 

5411 

1.8482 

5639 

1.7735 

35 

26 

4755 

2.1028 

4971 

2.0115 

5191 

1.9265 

5415 

1.8469 

5642 

1.7723 

34 

27 

4759 

2.1013 

4975 

2.0101 

5195 

1.9251 

5418 

1.8456 

5646 

1.7711 

33 

28 

4763 

2.0997 

4979 

2.0086 

5198 

1.9237 

5422 

1.8443 

5650 

1.7699 

32 

29 

4  766 

2.0981 

4982 

2.0072 

5202 

1.9223 

5426 

1.8430 

5654 

1.7687 

31 

30 

4770 

2.0965 

4986 

2.0057 

5206 

1.9210 

5430 

1.8418 

5658 

1.7675 

30 

31 

4773 

2.0950 

4989 

2.0042 

5209 

1.9196 

5433 

1.8405 

5662 

1.7663 

29 

32 

4777 

2.0934 

4993 

2.0028 

5213 

1.9183 

5437 

1.8392 

5665 

1.7651 

28 

33 

4780 

2.0918 

4997 

2.0013 

5217 

1.9169 

5441 

1.8379 

5669 

1.7639 

27 

34 

4784 

2.0903 

5000 

1.9999 

5220 

1.9155 

5445 

1.8367 

5673 

1.7627 

26 

35 

4788 

2.0887 

5004 

1.9984 

5224 

1.9142 

5448 

1.8354 

5677 

1.7615 

25 

36 

4791 

2.0872 

5008 

1.9970 

5228 

1.9128 

5452 

1.8341 

5681 

1.7603 

24 

37 

4795 

2.0856 

5011 

1.9955 

5232 

1.9115 

5456 

1.8329 

5685 

1.7591 

23 

38 

4798 

2.0840 

5015 

1.9941 

5235 

1.9101 

5460 

1.8316 

5688 

1.7579 

22 

39 

4802 

2.0825 

5019 

1.9926 

5239 

1.9088 

5464 

1.8303 

5692 

1.7567 

21 

40 

4806 

2.0809 

5022 

1.9912 

5243 

1.9074 

5467 

1.8291 

5696 

1.7556 

20 

41 

4809 

2.0794 

5026 

1.9897 

5246 

1.9061 

5471 

1.8278 

5700 

1.7544 

19 

42 

4813 

2.0778 

5029 

1.9883 

5250 

1.9047 

5475 

1.8265 

5704 

1.7532 

18 

43 

4816 

2.0763 

5033 

1.9868 

5254 

1.9034 

5479 

1.8253 

5708 

1.7520 

17 

44 

4820 

2.0748 

5037 

1.9854 

5258 

1.9020 

5482 

1.8240 

5712 

1.7508 

16 

45 

4823 

2.0732 

5040 

1.9840 

5261 

1.9007 

5486 

1.8228 

5715 

1.7496 

15 

46 

4827 

2.0717 

5044 

1.9825 

5265 

1.8993 

5490 

1.8215 

5719 

1.7485 

14 

47 

4831 

2.0701 

5048 

1.9811 

5269 

1.8980 

5494 

1.8202 

5723 

1.7473 

13 

48 

4834 

2.0686 

5051 

1.9797 

5272 

1.8967 

5498 

1.8190 

5727 

1.7461 

12 

49 

4838 

2.0671 

5055 

1.9782 

5276 

1.8953 

5501 

1.8177 

5731 

1.7449 

11 

50 

4841 

2.0655 

5059 

1.9768 

5280 

1.8940 

5505 

1.8165 

5735 

1.7437 

IO 

51 

4845 

2.0640 

5062 

1.9754 

5284 

1.8927 

5509 

1.8152 

5739 

1.7426 

9 

52 

4849 

2.0625 

5066 

1.9740 

5287 

1.8913 

5513 

1.8140 

5743 

1.7414 

8 

53 

4852 

2.0609 

5070 

1.9725 

5291 

1.8900 

5517 

1.8127 

5746 

1.7402 

7 

54 

4856 

2.0594 

5073 

1.9711 

5295 

1.8887 

5520 

1.8115 

5750 

1.7391 

6 

55 

4859 

2.0579 

5077 

1.9697 

5298 

1.8873 

5524 

1.8103 

5754 

1.7379 

5 

56 

4863 

2.0564 

5081 

1.9683 

5302 

1.8860 

5528 

1.8090 

5758 

1.7367 

4 

57 

4867 

2.0549 

5084 

1.9669 

5306 

1.8847 

5532 

1.8078 

5762 

1.7355 

3 

58 

4870 

2.0533 

5088 

1.9654 

5310 

1.8834 

5535 

1.8065 

5766 

1.7344 

2 

59 

4874 

2.0518 

5092 

1.9640 

5313 

1.8820 

5539 

1.8053 

5770 

1.7332 

1 

60 

4877 

2.0503 

5095 

1.9626 

5317 

1.8807 

5543 

1.8040 

5774 

1.7321 

O 

cot    tan 

64° 

cot   tan 

63° 

cot    tan 

62° 

cot 

tan 

cot   tan 

60° 

f 

f 

61° 

NATURAL  TANGENTS  AND 

COTANGENTS. 

67 

f 

i 

30° 

31° 

32° 

33° 

34° 

t 

tan 

cot 

tan   cot 

tan 

cot 

tan 

cot 

tan 

cot 

o 

5774 

1.7321 

6009  1.6643 

6249 

1.6003 

6494 

1.5399 

6745 

1.4826 

60 

1 

5777 

1.7309 

6013  1.6632 

6253 

1.5993 

6498 

1.5389 

6749 

1.4816 

59 

2 

5781 

1.7297 

6017  1.6621 

6257 

1.5983 

6502 

1.5379 

6754 

1.4807 

58 

3 

5785 

1.7286 

6020  1.6610 

6261 

1.5972 

6506 

1.5369 

6758 

1,4798 

57 

4 

5789 

1.7274 

6024  1.6599 

6265 

1.5962 

6511 

1.5359 

6762 

1.4788 

56 

5 

5793 

1.7262 

6028  1.6588 

6269 

1.5952 

6515 

1.5350 

6766 

1.4779 

55 

6 

5797 

1.7251 

6032  1.6577 

6273 

1.5941 

6519 

1.5340 

6771 

1.4770 

54 

7 

5801 

1.7239 

6036  1.6566 

6277 

1.5931 

6523 

1.5330 

6775 

1.4761 

53 

8 

5805 

1.7228 

6040  1.6555 

6281 

1.5921 

6527 

1.5320 

6779 

1.4751 

52 

9 

5808 

1.7216 

6044  1.6545 

6285 

1.5911 

6531 

1.5311 

6783 

1.4742 

51 

io 

5812 

1.7205 

6048  1.6534 

6289 

1.5900 

6536 

1.5301 

6787 

1.4733 

50 

11 

5816 

1.7193 

6052  1.6523 

6293 

1.5890 

6540 

1.5291 

6792 

1.4724 

49 

12 

5820 

1.7182 

6056  1.6512 

6297 

1.5880 

6544 

1.5282 

6796 

1.4715 

48 

13 

5824 

1.7170 

6060  1.6501 

6301 

1.5869 

6548 

1.5272 

6800 

1.4705 

47 

14 

5828 

1.7159 

6064  1.6490 

6305 

1.5859 

6552 

1.5262 

6805 

1.4696 

46 

15 

5832 

1.7147 

6068  1.6479 

6310 

1.5849 

6556 

1.5253 

6809 

1.4687 

45 

16 

5836 

1.7136 

6072  1.6469 

6314 

1.5839 

6560 

1.5243 

6813 

1.4678 

44 

17 

5840 

1.7124 

6076  1.6458 

6318 

1.5829 

6565 

1.5233 

6817 

1.4669 

43 

18 

5844 

1.7113 

6080  1.6447 

6322 

1.5818 

6569 

1.5224 

6822 

1.4659 

42 

19 

5847 

1.7102 

6084  1.6436 

6326 

1.5808 

6573 

1.5214 

6826 

1.4650 

41 

20 

5851 

1.7090 

6088  1.6426 

6330 

1.5798 

6577 

1.5204 

6830 

1.4641 

40 

21 

5855 

1.7079 

6092  1.6415 

6334 

1.5788 

6581 

1.5195 

6834 

1.4632 

39 

22 

5859 

1.7067 

6096  1.6404 

6338 

1.5778 

6585 

1.5185 

6839 

1.4623 

38 

23 

5863 

1.7056 

6100  1.6393 

6342 

1.5768 

6590 

1.5175 

6843 

1.4614 

37 

24 

5867 

1.7045 

6104  1.6383 

6346 

1.5757 

6594 

1.5166 

6847 

1.4605 

36 

25 

5871 

1.7033 

6108  1.6372 

6350 

1.5747 

6598 

1.5156 

6851 

1.4596 

35 

26 

5875 

1.7022 

6112  1.6361 

6354 

1.5737 

6602 

1.5147 

6856 

1.4586 

34 

27 

5879 

1.7011 

6116  1.6351 

6358 

1.5727 

6606 

1.5137 

6860 

1.4577 

33 

28 

5883 

1.6999 

6120  1.6340 

6363 

1.5717 

6610 

1.5127 

6864 

1.4568 

32 

29 

5887 

1.6988 

6124  1.6329 

6367 

1.5707 

6615 

1.5118 

6869 

1.4559 

31 

30 

5890 

1.6977 

6128  1.6319 

6371 

1.5697 

6619 

1.5108 

6873 

1.4550 

30 

31 

5894 

1.6965 

6132  1.6308 

6375 

1.5687 

6623 

1.5099 

6877 

1.4541 

29 

32 

5898 

1.6954 

6136  1.6297 

6379 

1.5677 

6627 

1.5089 

6881 

1.4532 

28 

33 

5902 

1.6943 

6140  1.6287 

6383 

1.5667 

6631 

1.5080 

6886 

1.4523  Z7 

34 

5906 

1.6932 

6144  1.6276 

6387 

1.5657 

6636 

1.5070 

6890 

1.4514 

26 

35 

5910 

1.6920 

6148  1.6265 

6391 

1.5647 

6640 

1.5061 

6894 

1.4505 

25 

36 

5914 

1.6909 

6152  1.6255 

6395 

1.5637 

6644 

1.5051 

6899 

1.4496 

24 

37 

5918 

1.6898 

6156  •  1.6244 

6399 

1.5627 

6648 

1.5042 

6903 

1.4487 

23 

38 

5922 

1.6887 

6160  1.6234 

6403 

1.5617 

6652 

1.5032 

6907 

1.4478 

22 

39 

5926 

1.6875 

6164  1.6223 

6408 

1.5607 

6657 

1.5023 

6911 

1.4469 

21 

40 

5930 

1.6864 

6168  1.6212 

6412 

1.5597 

6661 

1.5013 

6916 

1.4460 

20 

41 

5934 

1.6853 

6172  1.6202 

6416 

1.5587 

6665 

1.5004 

6920 

1.4451 

19 

42 

5938 

1.6842 

6176  1.6191 

6420 

1.5577 

6669 

1.4994 

6924 

1.4442 

18 

43 

5942 

1.6831 

6180  1.6181 

6424 

1.5567 

6673 

1.4985 

6929 

1.4433 

17 

44 

5945 

1.6820 

6184  1.6170 

6428 

1.5557 

6678 

1.4975 

o933 

1.4424 

16 

45 

5949 

1.6808 

6188  1.6160 

6432 

1.5547 

6682 

1.4966 

6937 

1.4415 

15 

46 

5953 

1.6797 

6192  1.6149 

6436 

1.5537 

6686 

1.4957 

6942 

1.4406 

14 

47 

5957 

1.6786 

6196  1.6139 

6440 

1.5527 

6690 

1.4947 

6946 

1.4397 

13 

48 

5961 

1.6775 

6200  1.6128 

6445 

1.5517 

6694 

1.4938 

6950 

1.4388 

12 

49 

5965 

1.6764 

6204  1.6118 

6449 

1.5507 

6699 

1.4928 

6954 

1.4379 

11 

SO 

5969 

1.6753 

6208  1.6107 

6453 

1.5497 

6703 

1.4919 

6959 

1.4370 

IO 

51 

5973 

1.6742 

6212  1.6097 

6457 

1.5487 

6707 

1.4910 

6963 

1.4361 

9 

52 

5977 

1.6731 

6216  1.6087 

6461 

1.5477 

6711 

1.4900 

6967 

1.4352 

8 

53 

5981 

1.6720 

6220  1.6076 

6465 

1.5468 

6716 

1.4891 

6972 

1.4344 

7 

54 

5985 

1.6709 

6224  1.6066 

6469 

1.5458 

6720 

1.4882 

6976 

1.4335 

6 

55 

5989 

1.6698 

6228  1.6055 

6473 

1.5448 

6724 

1.4872 

6980 

1.4326 

5 

56 

5993 

1.6687 

6233  1.6045 

6478 

1.5438 

6728 

1.4863 

6985 

1.4317 

4 

57 

5997 

1.6676 

6237  1.6034 

6482 

1.5428 

6732 

1.4854 

6989 

1.4308 

3 

58 

6001 

1.6665 

6241  1.6024 

6486 

1.5418 

6737 

1.4844 

6993 

1.4299 

2 

59 

6005 

1.6654 

6245  1.6014 

6490 

1.5408 

6741 

1.4835 

6998 

1.4290 

1 

60 

6009 

1.6643 

6249  1.6003 

6494 

1.5399 

6745 

1.4826 

7002 

1.4281 

O 

cot 

tan 

cot   tan 

cot 

tan 

cot 

tan 

cot 

tan 

f 

59° 

58° 

57° 

56° 

55° 

r 

68 

NATURAL  TANGENTS  AND 

COTANGENTS 

i. 

t 

35° 

36° 

37° 

38° 

39° 

t 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

O 

7002 

1.4281 

7265 

1.3764 

7536 

1.3270 

7813 

1.2799 

8098 

1.2349 

60 

l 

7006 

1.4273 

7270 

1.3755 

7540 

1.3262 

7818 

1.2792 

8103 

1.2342 

59 

2 

7011 

1.4264 

7274 

1.3747 

7545 

1.3254 

7822 

1.2784 

8107 

1.2334 

58 

3 

7015 

1.4255 

7279 

1.3739 

7549 

1.3246 

7827 

1.2776 

8112 

1.2327 

57 

4 

7019 

1.4246 

7283 

1.3730 

7554 

1.3238 

7832 

1.2769 

8117 

1.2320 

56 

5 

7024 

1.4237 

7288 

1.3722 

7558 

1.3230 

7836 

1.2761 

8122 

1.2312 

55 

6 

7028 

1.4229 

7292 

1.3713 

7563 

1.3222 

7841 

1.2753 

8127 

1.2305 

54 

7 

7032 

1.4220 

7297 

1.3705 

7568 

1.3214 

7846 

1.2746 

8132 

1.2298 

53 

8 

7037 

1.4211 

7301 

1.3697 

7572 

1.3206 

7850 

1.2738 

8136 

1.2290 

52 

9 

7041 

1.4202 

7306 

1.3688 

7577 

1.3198 

7855 

1.2731 

8141 

1.2283 

51 

10 

7046 

1.4193 

7310 

1.3680 

7581 

1.3190 

7860 

1.2723 

8146 

1.2276 

50 

11 

7050 

1.4185 

7314 

1.3672 

7586 

1.3182 

7865 

1.2715 

8151 

1.2268 

49 

12 

7054 

1.4176 

7319 

1.3663 

7590 

1.3175 

7869 

1.2708 

8156 

1.2261 

48 

13 

7059 

1.4167 

7323 

1.3655 

7595 

1.3167 

7874 

1.2700 

8161 

1.2254 

47 

14 

7063 

1.4158 

7328 

1.3647 

7600 

1.3159 

7879 

1.2693 

8165 

1.2247 

46 

15 

7067 

1.4150 

7332 

1.3638 

7604 

1.3151 

7883 

1.2685 

8170 

1.2239 

45 

16 

7072 

1.4141 

7337 

1.3630 

7609 

1.3143 

7888 

1.2677 

8175 

1.2232 

44 

17 

7076 

1.4132 

7341 

1.3622 

7613 

1.3135 

7893 

1.2670 

8180 

1.2225 

43 

18 

7080 

1.4124 

7346 

1.3613 

7618 

1.3127 

7898 

1.2662 

8185 

1.2218 

42 

19 

7085 

1.4115 

7350 

1.3605 

7623 

1.3119 

7902 

1.2655 

8190 

1.2210 

41 

20 

7089 

1.4106 

7355 

1.3597 

7627 

1.3111 

7907 

1.2647 

8195 

1.2203 

40 

21 

7094 

1.4097 

7359 

1.3588 

7632 

1.3103 

7912 

1.2640 

8199 

1.2196 

39 

22 

7098 

1.4089 

7364 

1.3580 

7636 

1.3095 

7916 

1.2632 

8204 

1.2189 

38 

23 

7102 

1.4080 

7368 

1.3572 

7641 

1.3087 

7921 

1.2624 

8209 

1.2181 

37 

24 

7107 

1.4071 

7373 

1.3564 

7646 

1.3079 

7926 

1.2617 

8214 

1.2174 

36 

25 

7111 

1.4063 

7377 

1.3555 

7650 

1.3072 

7931 

1.2609 

8219 

1.2167 

35 

26 

7115 

1.4054 

7382 

1.3547 

7655 

1.3064 

7935 

1.2602 

8224 

1.2160 

34 

27 

7120 

1.4045 

7386 

1.3539 

7659 

1.3056 

7940 

1.2594 

8229 

1.2153 

33 

28 

7124 

1.4037 

7391 

1.3531 

7664 

1.3048 

7945 

1.2587 

8234 

1.2145 

31 

29 

7129 

1.4028 

7395 

1.3522 

7669 

1.3040 

7950 

1.2579 

8238 

1.2138 

31 

30 

7133 

1.4019 

7400 

1.3514 

7673 

1.3032 

7954 

1.2572 

8243 

1.2131 

30 

31 

7137 

1.4011 

7404 

1.3506 

7678 

1.3024 

7959 

1.2564 

824S 

1.2124 

29 

32 

7142 

1.4002 

7409 

1.3498 

7683 

1.3017 

7964 

1.2557 

8253 

1.2117 

28 

33 

7146 

1.3994 

7413 

1.3490 

7687 

1.3009 

7969 

1.2549 

8258 

1.2109 

27 

34 

7151 

1.3985 

7418 

1.3481 

7692 

1.3001 

7973 

1.2542 

8263 

1.2102 

26 

35 

7155 

1.3976 

7422 

1.3473 

7696 

1.2993 

7978 

1.2534 

8268 

1.2095 

25 

36 

7159 

1.3968 

7427 

1.3465 

7701 

1.2985 

7983 

1.2527 

8273 

1.2088 

24 

37 

7164 

1.3959 

7431 

1.3457 

7706 

1.2977 

7988 

1.2519 

8278 

1.2081 

23 

38 

7168 

1.3951 

7436 

1.3449 

7710 

1.2970 

7992 

1.2512 

8283 

1.2074 

22 

39 

7173 

1.3942 

7440 

1.3440 

7715 

1.2962 

7997 

1.2504 

8287 

1.2066 

21 

40 

7177 

1.3934 

7445 

1.3432 

7720 

1.2954 

8002 

1.2497 

8292 

1.2059 

20 

41 

7181 

1.3925 

7449 

1.3424 

7724 

1.2946 

8007 

1.2489 

8297 

1.2052 

19 

42 

7186 

1.3916 

7454 

1.3416 

7729 

1.2938 

8012 

1.2482 

8302 

1.2045 

18 

43 

7190 

1.3908 

7458 

1.3408 

7734 

1.2931 

8016 

1.2475 

8307 

1.2038 

17 

44 

7195 

1.3899 

7463 

1.3400 

7738 

1.2923 

8021 

1.2467 

8312 

1.2031 

16 

45 

7199 

1.3891 

7467 

1.3392 

7743 

1.2915 

8026 

1.2460 

8317 

1.2024 

15 

46 

7203 

1.3882 

7472 

1.3384 

7747 

1.2907 

8031 

1.2452 

8322 

1.2017 

14 

47 

7208 

1.3874 

7476 

1.3375 

7752 

1.2900 

8035 

1.2445 

8327 

1.2009 

13 

48 

7212 

1.3865 

7481 

1.3367 

7757 

1.2892 

8040 

1.2437 

8332 

1.2002 

12 

49 

7217 

1.3857 

7485 

1.3359 

7761 

1.2884 

8045 

1.2430 

8337 

1.1995 

11 

50 

7221 

1.3848 

7490 

1.3351 

7766 

1.2876 

8050 

1.2423 

8342 

1.1988 

io 

51 

7226 

1.3840 

7495 

1.3343 

7771 

1.2869 

8055 

1.2415 

8346 

1.1981 

9 

52 

7230 

1.3831 

7499 

1.3335 

7775 

1.2861 

8059 

1.2408 

8351 

1.1974 

8 

53 

7234 

1.3823 

7504 

1.3327 

7780 

1.2853 

8064 

1.2401 

8356 

1.1967 

7 

54 

7239 

1.3814 

7508 

1.3319 

7785 

1.2846 

8069 

1.2393 

8361 

1.1960 

6 

55 

7243 

1.3806 

7513 

1.3311 

7789 

1.2838 

8074 

1.2386 

8366 

1.1953 

5 

56 

7248 

1.3798 

7517 

1.3303 

7794 

1.2830 

8079 

1.2378 

8371 

1.1946 

4 

57 

7252 

1.3789 

7522 

1.3295 

7799 

1.2822 

8083 

1.2371 

8376 

1.1939 

3 

58 

7257 

1.3781 

7526 

1.3287 

7803 

1.2815 

8088 

1.2364 

8381 

1.1932 

2 

59 

7261 

1.3772 

7531 

1.3278 

7808 

1.2807 

8093 

1.2356 

8386 

1.1925 

1 

60 

7265 

1.3764 

7536 

1.3270 

7813 

1.2799 

8098 

1.2349 

8391 

1.1918 

0 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

/ 

64° 

53° 

52° 

51° 

50° 

t 

NATURAL  TANGENTS  AND 

COTANGENTS. 

69 

; 

40° 

41° 

42° 

43° 

44° 

f 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

tan 

cot 

O 

8391 

1.1918 

8693 

1.1504 

9004 

1.1106 

9325 

1.0724 

9657 

1.0355 

60 

1 

8396 

1.1910 

8698 

1.1497 

9009 

1.1100 

9331 

1.0717 

9663 

1.0349 

59 

2 

8401 

1.1903 

8703 

1.1490 

9015 

1.1093 

9336 

1.0711 

9668 

1.0343 

58 

3 

8406 

1.1896 

8708 

1.1483 

9020 

1.1087 

9341 

1.0705 

9674 

1.0337 

57 

4 

8411 

1.1889 

8713 

1.1477 

9025 

1.1080 

9347 

1.0699 

9679 

1.0331 

56 

5 

8416 

1.1882 

8718 

1.1470 

9030 

1.1074 

9352 

1.0692 

9685 

1.0325 

55 

6 

8421 

1.1875 

8724 

1.1463 

9036 

1.1067 

9358 

1.0686 

9691 

1.0319 

54 

7 

8426 

1.1868 

8729 

1.1456 

9041 

1.1061 

9363 

1.0680 

9696 

1.0313 

53 

8 

8431 

1.1861 

8734 

1.1450 

9046 

1.1054 

9369 

1.0674 

9702 

1.0307 

52 

9 

8436 

1.1854 

8739 

1.1443 

9052 

1.1048 

9374 

1.0668 

9708 

1.0301 

51 

10 

8441 

1.1847 

8744 

1.1436 

9057 

1.1041 

9380 

1.0661 

9713 

1.0295 

50 

11 

8446 

1.1840 

8749 

1.1430 

9062 

1.1035 

9385 

1.0655 

9719 

1.0289 

49 

12 

8451 

1.1833 

8754 

1.1423 

9067 

1.1028 

9391 

1.0649 

9725 

1.0283 

48 

13 

8456 

1.1826 

8759 

1.1416 

9073 

1.1022 

9396 

1.0643 

9730 

1.0277 

47 

14 

8461 

1.1819 

8765 

1.1410 

9078 

1.1016 

9402 

1 .0637 

9736 

1.0271 

46 

15 

8466 

1.1812 

8770 

1.1403 

9083 

1.1009 

9407 

1.0630 

9742 

1.0265 

45 

16 

8471 

1.1806 

8775 

1.1396 

9089 

1.1003 

9413 

1.0624 

9747 

1.0259 

44 

17 

8476 

1.1799 

8780 

1.1389 

9094 

1.0996 

9418 

1.0618 

9753 

1.0253 

43 

18 

8481 

1.1792 

8785 

1.1383 

9099 

1.0990 

9424 

1.0612 

9759 

1.0247 

42 

19 

8486 

1.1785 

8790 

1.1376 

9105 

1.0983 

9429 

1.0606 

9764 

1.0241 

41 

20 

8491 

1.1778 

8796 

1.1369 

9110 

1.0977 

9435 

1.0599 

9770 

1.0235 

40 

21 

8496 

1.1771 

8801 

1.1363 

9115 

1.0971 

9440 

1.0593 

9776 

1.0230 

39 

22 

8501 

1.1764 

8806 

1.1356 

9121 

1.0964 

9446 

1.0587 

9781 

1.0224 

38 

23 

8506 

1.1757 

8811 

1.1349 

9126 

1.0958 

9451 

1.0581 

9787 

1.0218 

37 

24 

8511 

1.1750 

8816 

1.1343 

9131 

1.0951 

9457 

1.0575 

9793 

1.0212 

36 

^5 

8516 

1.1743 

8821 

1.1336 

9137 

1.0945 

9462 

1.0569 

9798 

1.0206 

35 

26 

8521 

1.1736 

8827 

1.1329 

9142 

1.0939 

9468 

1.0562 

9804 

1.0200 

34 

27 

8526 

1.1729 

8832 

1.1323 

9147 

1.0932 

9473 

1.0556 

9810 

1.0194 

33 

28 

8531 

1.1722 

8837 

1.1316 

9153 

1.0926 

9479 

1.0550 

9816 

1.0188 

32 

29 

8536 

1.1715 

8842 

1.1310 

9158 

1.0919 

9484 

1.0544 

9821 

1.0182 

31 

30 

8541 

1.1708 

8847 

1.1303 

9163 

1.0913 

9490 

1.0538 

9827 

1.0176 

30 

31 

8546 

1.1702 

8852 

1.1296 

9169 

1.0907 

9495 

1.0532 

9833 

1.0170 

29 

32 

8551 

1.1695 

8858 

1.1290 

9174 

1.0900 

9501 

1.0526 

9838 

1.0164 

28 

33 

8556 

1.1688 

8863 

1.1283 

9179 

1.0894 

9506 

1.0519 

9844 

1.0158 

27 

34 

8561 

1.1681 

8868 

1.1276 

9185 

1.0888 

9512 

1.0513 

9850 

1.0152 

26 

35 

8566 

1.1674 

8873 

1.1270 

9190 

1.0881 

9517 

1.0507 

9856 

1.0147 

25 

36 

8571 

1.1667 

8878 

1.1263 

9195 

1.0875 

9523 

1.0501 

9861 

1.0141 

24 

37 

8576 

1.1660 

8884 

1.1257 

9201 

1.0869 

9528 

1.0495 

9867 

1.0135 

23 

38 

8581 

1.1653 

8889 

1.1250 

9206 

1.0862 

9534 

1.0489 

9873 

1.0129 

22 

39 

8586 

1.1647 

8894 

1.1243 

9212 

1.0856 

9540 

1.0483 

9879 

1.0123 

21 

40 

8591 

1.1640 

8899 

1.1237 

9217 

1.0850 

9545 

1.0477 

9884 

1.0117 

20 

41 

8596 

1.1633 

8904 

1.1230 

9222 

1.0843 

9551 

1.0470 

9890 

1.0111 

19 

42 

8601 

1.1626 

8910 

1.1224 

9228 

1.0837 

9556 

1.0464 

9896 

1.0105 

18 

43 

8606 

1.1619 

8915 

1.1217 

9233 

1.0831 

9562 

1.0458 

9902 

1.0099 

17 

44 

8611 

1.1612 

8920 

1.1211 

9239 

1.0824 

9567 

1.0452 

9907 

1.0094 

16 

45 

8617 

1.1606 

8925 

1.1204 

9244 

1.0818 

9573 

1.0446 

9913 

1.0088 

15 

46 

8622 

1.1599 

8931 

1.1197 

9249 

1.0812 

9578 

1.0440 

9919 

1.0082 

14 

47 

8627 

1.1592 

8936 

1.1191 

9255 

1.0805 

9584 

1.0434 

9925 

1.0076 

13 

48 

8632 

1.1585 

8941 

1.1184 

9260 

1.0799 

9590 

1.0428 

9930 

1.0070 

12 

49 

8637 

1.1578 

8946 

1.1178 

9266 

1.0793 

9595 

1.0422 

9936 

1.0064 

11 

50 

8642 

1.1571 

8952 

1.1171 

9271 

1.0786 

9601 

1.0416 

9942 

1.0058 

10 

51 

8647 

1.1565 

8957 

1.1165 

9276 

1.0780 

9606 

1.0410 

9948 

1.0052 

9 

52 

8652 

1.1558 

8962 

1.1158 

9282 

1.0774 

9612 

1.0404 

9954 

1.0047 

8 

53 

8657 

1.1551 

8967 

1.1152 

9287 

1.0768 

9618 

1.0398 

9959 

1.0041 

7 

54 

8662 

1.1544 

8972 

1.1145 

9293 

1.0761 

9623 

1.0392 

9965 

1.0035 

6 

55 

8667 

1.1538 

8978 

1.1139 

9298 

1.0755 

9629 

1.0385 

9971 

1.0029 

5 

56 

8672 

1.1531 

8983 

1.1132 

9303 

1.0749 

9634 

1.0379 

9977 

1.0023 

4 

57 

8678 

1.1524 

8988 

1.1126 

9309 

1.0742 

9640 

1.0373 

9983 

1.0017 

3 

58 

8683 

1.1517 

8994 

1.1119 

9314 

1.0736 

9646 

1.0367 

9988 

1.0012 

2 

59 

8688 

1.1510 

8999 

1.1113 

9320 

1.0730 

9651 

1.0361 

9994 

1.0006 

1 

60 

8693 

1.1504 

9004 

1.1106 

9325 

1.0724 

9657 

1.0355 

1000 

1.0000 

O 

/ 

cot   tan 

49° 

cot   tan 
48° 

cot   tan 

47° 

cot 

tan 

cot   tan 
45° 

t 

46° 

A  TABLE   OF  THE  ANGLES 

Which  every  Point  and  Quarter  Point  of  the  Compass  makes  with  the  Meridian. 


North. 

Points. 

o-3I 

O        /       (/ 

Points. 

South. 

N.  by  E. 

N.  by  W. 

2  48  45 

5  37  30 

8  26  15 

11  15    0 

o-V4 
o-v2 
0-34 

S.  by  E. 

S.  by  W. 

N.N.E. 

N.N.W. 

14    3  45 
16  52  30 
19  41  15 
22  30    0 

S.S.E. 

S.S.W. 

N.E.  by  N. 

N.W.  by  N. 

3 

3 

25  18  45 
28     7  30 
30  56  15 
33  45    0 

S.E.  by  S. 

S.W.  by  S. 

N.E. 

N.W. 

3-V4 
3-% 
3-% 
4 

36  33  45 
39  22  30 
42  11  15 
45    0    0 

3-V* 

3-% 
3-% 
4 

S.E. 

S.W. 

N.E.  by  E. 

N.W.  by  W. 

Sit 

5 

47  48  46 
50  37  30 
53  26  15 
56  15    0 

4 -V4 

4-? 

5 

S.E.  by  E. 

S.W.  by  W. 

E.N.E. 

W.N.W. 

6 

59    3  45 
61  62  30 
64  41  15 

07  30     0 

5-% 
5-% 
5-sJ 
6 

E.S.E. 

W.S.W. 

E.  by  N. 

W.  by  N. 

j:f 

70  18  45 
73    7  30 
75  56  15 
78  45    0 

6-% 
6-V0 
6-% 
7 

E.  by  S. 

W.  by  S. 

East. 

West. 

P 

81  33  46 
84  22  30 
87  11  15 
90    0    0 

East. 

West. 

o 


V 


w     <*  BOOK 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


DEC  24  1^7      REC'D  l.q 
'MAY  4 


12 


190 
MAY2  2 1953  LU 


<xtf* 


•^ 


8  Je'59ESf 
REC'D  LD 

MAY  25  1959 


LD  21-100m-9,'47(A5702sl6)476 


^ 


m 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


